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Periodic groups with given element orders V. D. Mazurov Sobolev Institute of Mathematics Siberian Branch of Russian Academy of Sciences Novosibirsk, November 2012 Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 1 / 19 A group G is said to be periodic if the order of every element of G is finite, i.e., for every g ∈ G, there exist a natural n (depending on g) such that g n = 1. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 2 / 19 A group G is said to be periodic if the order of every element of G is finite, i.e., for every g ∈ G, there exist a natural n (depending on g) such that g n = 1. Every finite group is periodic, but there exist many periodic groups which are infinite. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 2 / 19 A group G is said to be periodic if the order of every element of G is finite, i.e., for every g ∈ G, there exist a natural n (depending on g) such that g n = 1. Every finite group is periodic, but there exist many periodic groups which are infinite. If there exists a common natural n such that xn = 1 for every x ∈ G then the smallest such n is called the exponent of G. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 2 / 19 A group G is said to be periodic if the order of every element of G is finite, i.e., for every g ∈ G, there exist a natural n (depending on g) such that g n = 1. Every finite group is periodic, but there exist many periodic groups which are infinite. If there exists a common natural n such that xn = 1 for every x ∈ G then the smallest such n is called the exponent of G. A group G is said to be locally finite if every finite subset of G generates a finite subgroup. Every finite group is, of course, locally finite, and every locally finite group is periodic, but not every periodic group is locally finite. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 2 / 19 A history of research of periodic groups with prescribed element orders starts with the famous work of W. Burnside [On an unsettled question in the theory of discontinuous groups // Quart. J. Pure Appl. Math.—1902.—Vol. 37.—P. 230–238] where he firstly stated his known problems. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 3 / 19 A history of research of periodic groups with prescribed element orders starts with the famous work of W. Burnside [On an unsettled question in the theory of discontinuous groups // Quart. J. Pure Appl. Math.—1902.—Vol. 37.—P. 230–238] where he firstly stated his known problems. In particular, he posed a question on conditions which guarantee that a group of given bounded exponent is locally finite. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 3 / 19 A history of research of periodic groups with prescribed element orders starts with the famous work of W. Burnside [On an unsettled question in the theory of discontinuous groups // Quart. J. Pure Appl. Math.—1902.—Vol. 37.—P. 230–238] where he firstly stated his known problems. In particular, he posed a question on conditions which guarantee that a group of given bounded exponent is locally finite. Burnside noted an obvious fact that a group of exponent 2 is locally finite (in fact, it is commutative), and showed that this is also true for groups of exponent 3 Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 3 / 19 A history of research of periodic groups with prescribed element orders starts with the famous work of W. Burnside [On an unsettled question in the theory of discontinuous groups // Quart. J. Pure Appl. Math.—1902.—Vol. 37.—P. 230–238] where he firstly stated his known problems. In particular, he posed a question on conditions which guarantee that a group of given bounded exponent is locally finite. Burnside noted an obvious fact that a group of exponent 2 is locally finite (in fact, it is commutative), and showed that this is also true for groups of exponent 3 He also proved that in groups of exponent 3 every couple of conjugate elements commute, i. e. in such groups the 2-Engel identity [[y, x], x] = 1 holds where [x, y] = x−1 y −1 xy. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 3 / 19 A history of research of periodic groups with prescribed element orders starts with the famous work of W. Burnside [On an unsettled question in the theory of discontinuous groups // Quart. J. Pure Appl. Math.—1902.—Vol. 37.—P. 230–238] where he firstly stated his known problems. In particular, he posed a question on conditions which guarantee that a group of given bounded exponent is locally finite. Burnside noted an obvious fact that a group of exponent 2 is locally finite (in fact, it is commutative), and showed that this is also true for groups of exponent 3 He also proved that in groups of exponent 3 every couple of conjugate elements commute, i. e. in such groups the 2-Engel identity [[y, x], x] = 1 holds where [x, y] = x−1 y −1 xy. In a sequel [Burnside W. On groups in which every two conjugate operations are permutable // Proc. London Math. Soc.—1902.—Vol. 35.—P. 28–37] of this work Burnside proved that every 2-Engel group satisfies the identities [[x, y], z] = [[y, z], x] and [[x, y], z]3 = 1, thus it is nilpotent of nilpotency class 2 (this means that the identity [[x, y], z] = 1 holds), in the case when it has no elements of Novosibirsk, November 2012 3 / V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 19 order 3. Apparently, C. Hopkins [Hopkins C. Finite groups in which conjugate operations are commutative // Amer. J. Math.—1929.—Vol. 51.—P.35–41.] was first to prove that a 2-Engel group is nilpotent of class 3, in particular such are groups of exponent 3. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 4 / 19 Apparently, C. Hopkins [Hopkins C. Finite groups in which conjugate operations are commutative // Amer. J. Math.—1929.—Vol. 51.—P.35–41.] was first to prove that a 2-Engel group is nilpotent of class 3, in particular such are groups of exponent 3. This result is usually ascribed to F. Levi [Levi F. V. Groups in which the commutator operations satisfy certain algebraic conditions // J. Indian Math. Soc.—1942.—Vol. 6.—P.166–170], although he published his work 13 years later than Hopkins. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 4 / 19 Apparently, C. Hopkins [Hopkins C. Finite groups in which conjugate operations are commutative // Amer. J. Math.—1929.—Vol. 51.—P.35–41.] was first to prove that a 2-Engel group is nilpotent of class 3, in particular such are groups of exponent 3. This result is usually ascribed to F. Levi [Levi F. V. Groups in which the commutator operations satisfy certain algebraic conditions // J. Indian Math. Soc.—1942.—Vol. 6.—P.166–170], although he published his work 13 years later than Hopkins. In 1932 Levi and B. van der Waerden [Levi F., van der Waerden B. Über eine besondere Klasse von Gruppen // Abh. Math. Semin., Hamburg Univ.—1932.—Vol. 9.—P. 157–158.] repeated Burnside’s result that a group of exponent 3 is 2-Engel. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 4 / 19 Apparently, C. Hopkins [Hopkins C. Finite groups in which conjugate operations are commutative // Amer. J. Math.—1929.—Vol. 51.—P.35–41.] was first to prove that a 2-Engel group is nilpotent of class 3, in particular such are groups of exponent 3. This result is usually ascribed to F. Levi [Levi F. V. Groups in which the commutator operations satisfy certain algebraic conditions // J. Indian Math. Soc.—1942.—Vol. 6.—P.166–170], although he published his work 13 years later than Hopkins. In 1932 Levi and B. van der Waerden [Levi F., van der Waerden B. Über eine besondere Klasse von Gruppen // Abh. Math. Semin., Hamburg Univ.—1932.—Vol. 9.—P. 157–158.] repeated Burnside’s result that a group of exponent 3 is 2-Engel. They also proved that the order of a group of exponent 3 with d generators is at most 3k , where k = (6d + 3d(d − 1) + d(d − 1)(d − 2))/6. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 4 / 19 For convenience in listing further results which are tied with our topic, we set for a group G the following notations: Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 5 / 19 For convenience in listing further results which are tied with our topic, we set for a group G the following notations: ω(G) = {n | G has an element of order n} is a spectrum of a group G; Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 5 / 19 For convenience in listing further results which are tied with our topic, we set for a group G the following notations: ω(G) = {n | G has an element of order n} is a spectrum of a group G; µ(G) is a set of maximal by divisibility elements of spectrum of G. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 5 / 19 For convenience in listing further results which are tied with our topic, we set for a group G the following notations: ω(G) = {n | G has an element of order n} is a spectrum of a group G; µ(G) is a set of maximal by divisibility elements of spectrum of G. A group of period n is a group where the identity xn = 1 holds, and a group of nilpotency class n is a group in which the equality −1 [x1 , x2 , . . . , xn+1 ] = 1 holds, where [x1 , x2 ] = x−1 1 x2 x1 x2 and [x1 , x2 , . . . , xn+1 ] = [[x1 , x2 , . . . , xn ], xn+1 ] for n > 2. In this sense 2-group of exponent 2 is nilpotent of class 2 and of period 36. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 5 / 19 For convenience in listing further results which are tied with our topic, we set for a group G the following notations: ω(G) = {n | G has an element of order n} is a spectrum of a group G; µ(G) is a set of maximal by divisibility elements of spectrum of G. A group of period n is a group where the identity xn = 1 holds, and a group of nilpotency class n is a group in which the equality −1 [x1 , x2 , . . . , xn+1 ] = 1 holds, where [x1 , x2 ] = x−1 1 x2 x1 x2 and [x1 , x2 , . . . , xn+1 ] = [[x1 , x2 , . . . , xn ], xn+1 ] for n > 2. In this sense 2-group of exponent 2 is nilpotent of class 2 and of period 36. We say that a group G acts freely on a nontrivial group V , if v g 6= v for any nontrivial v ∈ V and g ∈ G. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 5 / 19 Groups with prescribed spectrum Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 6 / 19 Groups with prescribed spectrum In 1937 B. Neumann [Neumann B. H. Groups whose elements have bounded orders // J. London Math. Soc.—1937.—Vol. 12.—P. 195–198] saw it natural to investigate groups with prescribed spectrum. In particular, he proved that a group G with µ(G) = {2, 3} is locally finite and soluble of length 2. More precisely, such group is an extension of an elementary abelian p-group V by a cyclic q-group acting freely on V . Here {p, q} = {2, 3}. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 6 / 19 Groups with prescribed spectrum In 1937 B. Neumann [Neumann B. H. Groups whose elements have bounded orders // J. London Math. Soc.—1937.—Vol. 12.—P. 195–198] saw it natural to investigate groups with prescribed spectrum. In particular, he proved that a group G with µ(G) = {2, 3} is locally finite and soluble of length 2. More precisely, such group is an extension of an elementary abelian p-group V by a cyclic q-group acting freely on V . Here {p, q} = {2, 3}. The fact that a group G with µ(G) ⊆ {3, 4} is locally finite was proved by I. N. Sanov in 1940 [In Russian: Solution of Burnside problem for period 4 // Sci. Notes of Leingrad Stats University. Math. Ser. —1940.—Vol. 10, no. 55.—P. 166–170]. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 6 / 19 Groups with prescribed spectrum In 1937 B. Neumann [Neumann B. H. Groups whose elements have bounded orders // J. London Math. Soc.—1937.—Vol. 12.—P. 195–198] saw it natural to investigate groups with prescribed spectrum. In particular, he proved that a group G with µ(G) = {2, 3} is locally finite and soluble of length 2. More precisely, such group is an extension of an elementary abelian p-group V by a cyclic q-group acting freely on V . Here {p, q} = {2, 3}. The fact that a group G with µ(G) ⊆ {3, 4} is locally finite was proved by I. N. Sanov in 1940 [In Russian: Solution of Burnside problem for period 4 // Sci. Notes of Leingrad Stats University. Math. Ser. —1940.—Vol. 10, no. 55.—P. 166–170]. Later D. V. Lytkina [In Russian: Structure of a group whose element orders are at most 4 // Siberian Math. J.—2007.—Vol. 48, no. 2.—P. 353–358] described the structure of a group G with µ(G) = {3, 4}. She proved that G is soluble of length at most 3 and only one of the following cases is possible: Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 6 / 19 (a) G = V Q, where V is a non-trivial normal elementary abelian 3-subgroup, Q is a 2-group, acting freely on V , and is isomorphic either to a cyclic group of order 4, or to quaternion group of order 8; (b) G = T hai, where T is a normal nilpotent 2-subgroup of nilpotency class 2, and the order of a equals 3; (c) G = T S, where T is an elementary abelian normal 2-subgroup, and S is isomorphic to a symmetric group of degree 3. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 7 / 19 (a) G = V Q, where V is a non-trivial normal elementary abelian 3-subgroup, Q is a 2-group, acting freely on V , and is isomorphic either to a cyclic group of order 4, or to quaternion group of order 8; (b) G = T hai, where T is a normal nilpotent 2-subgroup of nilpotency class 2, and the order of a equals 3; (c) G = T S, where T is an elementary abelian normal 2-subgroup, and S is isomorphic to a symmetric group of degree 3. For a finite group of period 4 its solubility length grows unlimited with the number of generators growth [Razmyslov Yu. P. Hall-Higman’s Problem // Proc. Academy of Sci. USSR. Math. Ser.—1978.—Vol. 42, no. 4.—P. 833–847.] Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 7 / 19 Groups of period 6 In 1956 a famous article by P. Hall and G. Higman [Hall P., Higman G. [On the p-length of p-soluble groups and reduction theorems for Burnside’s problem // Proc. London Math. Soc.—1956.—Vol. 6, № 3.—P. 1–42] was issued, and it equipped mathematicians with new powerfull tools for exploration of finite groups. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 8 / 19 Groups of period 6 In 1956 a famous article by P. Hall and G. Higman [Hall P., Higman G. [On the p-length of p-soluble groups and reduction theorems for Burnside’s problem // Proc. London Math. Soc.—1956.—Vol. 6, № 3.—P. 1–42] was issued, and it equipped mathematicians with new powerfull tools for exploration of finite groups. In particular, it moved M. Hall to write down his work [Hall M. Solution of the Burnside problem for exponent six // Illinois J. Math.—1958.—Vol. 2.—P. 764–786], where he proved that a group of period 6 is locally finite. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 8 / 19 Groups of period 6 In 1956 a famous article by P. Hall and G. Higman [Hall P., Higman G. [On the p-length of p-soluble groups and reduction theorems for Burnside’s problem // Proc. London Math. Soc.—1956.—Vol. 6, № 3.—P. 1–42] was issued, and it equipped mathematicians with new powerfull tools for exploration of finite groups. In particular, it moved M. Hall to write down his work [Hall M. Solution of the Burnside problem for exponent six // Illinois J. Math.—1958.—Vol. 2.—P. 764–786], where he proved that a group of period 6 is locally finite. This result together with results from the article by Hall and Higman imply that all such groups are soluble of length at most 4, and the order of a d-generated group of period 6 is at most 2a 3b+b(b−1)/2+b(b−1)(b−2)/6 , where a = 1 + (d − 1)3d+d(d−1)/2+d(d−1)(d−2)/6 , b = 1 + (d − 1)2d , and this bound is exact.Periodic groups with given element orders Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) 8 / 19 Note that later M. Newman [Newman M. F. Groups of exponent six // Computational group theory (Durham, 1982), London: Academic Press.—1984.—P. 39–41] shortened essentially Hall’s proof, reducing it to some statement which he checked with the help of a computer. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 9 / 19 Note that later M. Newman [Newman M. F. Groups of exponent six // Computational group theory (Durham, 1982), London: Academic Press.—1984.—P. 39–41] shortened essentially Hall’s proof, reducing it to some statement which he checked with the help of a computer. I. G. Lysenok [In Russian: Lysenok I. G. A proof of M. Hall on the finiteness of groups B(m, 6) // Math. Notes—1987.—Vol. 41, no. 3.—P. 422–428] in 1987 delivered Mewman’s proof of computer computations. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 9 / 19 Negative solution In 1959 P. S. Novikov [In Russian: On periodic groups // Reports of Academy Sciences of USSR—1959.—Vol. 127.—P. 749–752] announced the existence of infinite, finitely generated periodic groups of finite period. On the ground of this note Novikov and S. I. Adian in 1968 wrote a big article [On infinite periodic groupI // Izvestiya: Mathematics—1968.—Vol. 32—P. 212–244; 251-524; 709-731] with a proof that there exists a m-generated group of period n for every m > 2 and every odd n > 4381. In the book of Adian [In Russian: Adian S. I. Burnside Problem and Identities in Groups.—Moscow: Nauka, 1975], the bound n > 4381 was decreased to n > 665. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 10 / 19 Negative solution In 1959 P. S. Novikov [In Russian: On periodic groups // Reports of Academy Sciences of USSR—1959.—Vol. 127.—P. 749–752] announced the existence of infinite, finitely generated periodic groups of finite period. On the ground of this note Novikov and S. I. Adian in 1968 wrote a big article [On infinite periodic groupI // Izvestiya: Mathematics—1968.—Vol. 32—P. 212–244; 251-524; 709-731] with a proof that there exists a m-generated group of period n for every m > 2 and every odd n > 4381. In the book of Adian [In Russian: Adian S. I. Burnside Problem and Identities in Groups.—Moscow: Nauka, 1975], the bound n > 4381 was decreased to n > 665. Existance of non locally finite groups of finite period of the shape 2t was announced in 1992 independantly by S. V. Ivanov and Lysenok. Their works [Ivanov S. V. The free Burnside groups of sufficiently large exponents // Internat. J. Algebra Comput.—1994.—V. 4.—P. 3–308] and [In Russian: Lysenok I. G. Infinite Burnside groups of even period // Izvestiya: Mathematics—1996.—Vol. 60, no 3.—P. 3–224] with full proofs were issued in 1994 and 1996 correspondingly. In particular, Lysenok’s work contains a proof of existance of infinite m-generated groups of period n for every m > 2 and n > 8000. Novosibirsk, November 2012 10 / V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 19 Again groups with prescribed spectrum In 1986 W. Shi [In Chinese: Shi W. A characteristic property of A5 // J. Southwest-China Teachers Univ.—1986.—V. 3.—P. 11–14] proved that alternating group A5 ' L2 (4) is the only group with spectrum {1, 2, 3, 5}, which opened a broad way for getting different characterisations of finite groups that use the idea of a spectrum. We shall not refer to this vast topic. We note only that later Shi proved (among many other results in this direction) that every simple group L2 (2m ) is recognizable by its spectrum in the class of finite groups. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 11 / 19 Again groups with prescribed spectrum In 1986 W. Shi [In Chinese: Shi W. A characteristic property of A5 // J. Southwest-China Teachers Univ.—1986.—V. 3.—P. 11–14] proved that alternating group A5 ' L2 (4) is the only group with spectrum {1, 2, 3, 5}, which opened a broad way for getting different characterisations of finite groups that use the idea of a spectrum. We shall not refer to this vast topic. We note only that later Shi proved (among many other results in this direction) that every simple group L2 (2m ) is recognizable by its spectrum in the class of finite groups. In 1999 Zhurtov and V. D. Mazurov [In Russian: Zhurtov A. Kh., Mazurov V. D. Recognition of finite simple groups L2 (2m ) in the class of all groups // Siberian Math. J.—1999.—Vol. 40, no. 1.—P. 75–78] showed that this is true even if we do not require finiteness of a group: µ(G) = {2, 2m − 1, 2m + 1} if and only if G ' L2 (2m ). Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 11 / 19 In the spectrum of a group L2 (Q), where Q is an arbitrary locally finite field of characteristic 2, the only even number is 2, and in 2000 Mazurov [In Russian: Mazurov V. D. Infinite groups with abelian centralizers of involutions // Algebra and Logic—2000.—Vol. 39, no. 1.—P. 74–86.] found the general shape of a group G with a spectrum of the type {2} ∪ ω 0 , where ω 0 contains only odd numbers. He showed that for G one of the following statements is true: Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 12 / 19 In the spectrum of a group L2 (Q), where Q is an arbitrary locally finite field of characteristic 2, the only even number is 2, and in 2000 Mazurov [In Russian: Mazurov V. D. Infinite groups with abelian centralizers of involutions // Algebra and Logic—2000.—Vol. 39, no. 1.—P. 74–86.] found the general shape of a group G with a spectrum of the type {2} ∪ ω 0 , where ω 0 contains only odd numbers. He showed that for G one of the following statements is true: (a) G is an extension of an abelian group A by a group hti of order 2, and at = a−1 for every a ∈ A. (b) G is an extension of an elementary abelian 2-group A by a group without involutions, acting freely on A by conjugation in G. (c) G ' L2 (Q) for sufficient locally finite field Q of characteristic 2. Groups in items (a) and (c) are locally finite. There exist groups from item (b), which are not locally finite. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 12 / 19 In the spectrum of a group L2 (Q), where Q is an arbitrary locally finite field of characteristic 2, the only even number is 2, and in 2000 Mazurov [In Russian: Mazurov V. D. Infinite groups with abelian centralizers of involutions // Algebra and Logic—2000.—Vol. 39, no. 1.—P. 74–86.] found the general shape of a group G with a spectrum of the type {2} ∪ ω 0 , where ω 0 contains only odd numbers. He showed that for G one of the following statements is true: (a) G is an extension of an abelian group A by a group hti of order 2, and at = a−1 for every a ∈ A. (b) G is an extension of an elementary abelian 2-group A by a group without involutions, acting freely on A by conjugation in G. (c) G ' L2 (Q) for sufficient locally finite field Q of characteristic 2. Groups in items (a) and (c) are locally finite. There exist groups from item (b), which are not locally finite. Thus, existence of a non locally finite group G with such a spectrum strictly depends on existence of a group with the spectrum ω 0 , which can act freely on an elementary abelian 2-group. In particular, if ω 0 is equal to one of the sets {1, 3, 9} or {1, 3, 5, 9, 15}, then a group is locally finite [Zhurtov A. Kh., Mazurov V. D. Local finiteness of som groups with given element orders // Vladikavkaz. Math. J.—2009.—Vol. 11, Novosibirsk, November 2012 12 / no. 4.—P. 11–15.] V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 19 In two papers [Gupta N. D., Mazurov V. D. On groups with small orders of elements // Bull. Austral. Math. Soc.—1999.—V. 60.—P. 197–205.] and and [Mazurov V. D. On groups of period 60 with given orders of elements // Algebra and Logic—2000.—Vol. 39, no. 3.—P. 329–346] N. Gupta and Mazurov considered groups whose spectrum is a subset of a set {1, 2, 3, 4, 5}. All of them are locally finite with possible exception of groups of period 5. More precisely, Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 13 / 19 In two papers [Gupta N. D., Mazurov V. D. On groups with small orders of elements // Bull. Austral. Math. Soc.—1999.—V. 60.—P. 197–205.] and and [Mazurov V. D. On groups of period 60 with given orders of elements // Algebra and Logic—2000.—Vol. 39, no. 3.—P. 329–346] N. Gupta and Mazurov considered groups whose spectrum is a subset of a set {1, 2, 3, 4, 5}. All of them are locally finite with possible exception of groups of period 5. More precisely, • • • If µ(G) = {3, 5}, then either G = F T , where F is a nilpotent of class 2 normal 5-subgroup, and |T | = 3, or G is an extension of a nilpotent of class 3 3-group by a group of order 5. If µ(G) = {4, 5}, then on of the following statements holds. (a) G = T D, where T is a normal elementary abelian 2-group, and D is a nonalbelian group of order 10. (b) G = F T , where F is an elementary abelian normal 5-subgroup, and T is isomorphic to a subgroup of quaternion group of order 8. (c) G = T F , where T is a nilpotent of class 6 normal 2-subgroup, and F is a group of order 5. If µ(G) = {3, 4, 5}, then G is locally finite and is either isomorphic to A6 , or G = V C, where V is a nontrivial elementary abelian normal 2-subgroup, and C ' A5 . Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 13 / 19 In 2007 Lytkina and Kuznetsov [Lytkina D. V., Kuznetsov A. A. Recognizability by spectrum of the group L2 (7) in the class of all groups // Sib. Electronic Math. Reports.—2007.—V. 4.—P. 300–303] characterised the simple group L2 (7) by spectrum in the class of all groups. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 14 / 19 In 2007 Lytkina and Kuznetsov [Lytkina D. V., Kuznetsov A. A. Recognizability by spectrum of the group L2 (7) in the class of all groups // Sib. Electronic Math. Reports.—2007.—V. 4.—P. 300–303] characterised the simple group L2 (7) by spectrum in the class of all groups. • If µ(G) = {3, 4, 7}, then G ' L2 (7). Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 14 / 19 In 2009 Mazurov and A. S. Mamontov [In Russian: Mazurov V. D., Mamontov A. S. Periodic groups with elements of small orders // Siberian Math. J.—2009.—Vol. 50, no. 2.—P. 397–404] proved the following result. • Suppose µ(G) = {5, 6}. Then G is a soluble locally finite group and one of the following statements is true: (a) G is an extensiion of an elementary abelian 5-group by a cyclic group of order 6; (b) G is an extension of a nilpotent of class 3 3-group by a dihedral group of order 10; (c) G is an extension of a direct product of a nilpotent of class 3 3-group and elementary abelian 2-group by a group of order 5. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 15 / 19 In 2009 Mazurov and A. S. Mamontov [In Russian: Mazurov V. D., Mamontov A. S. Periodic groups with elements of small orders // Siberian Math. J.—2009.—Vol. 50, no. 2.—P. 397–404] proved the following result. • Suppose µ(G) = {5, 6}. Then G is a soluble locally finite group and one of the following statements is true: (a) G is an extensiion of an elementary abelian 5-group by a cyclic group of order 6; (b) G is an extension of a nilpotent of class 3 3-group by a dihedral group of order 10; (c) G is an extension of a direct product of a nilpotent of class 3 3-group and elementary abelian 2-group by a group of order 5. In 2010 Mazurov [In Russian: Mazurov V. D. On groups of period 24 // Algebra and Logic—2010.—Vol. 49, no. 6.—P. 766–781] showed that a group G with µ(G) = {3, 8} is locally finite and for G one of the following statements holds: (a) G = V Q, where V is a nontrivial normal elementary abelian 3-subgroup, Q is a 2-group acting freely on V and isomorphic either to a cyclic group of order 8, or to a quaternion group of order 16; (b) G = T hai, where T is a normal nilpotent of class 2 2-subgroup, and the order of a euqals 3; Novosibirsk, November 2012 15 / V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 19 New (yet not published) results (E. Jabara, D. V. Lytkina) Suppose G is a group of period 36 which does not possess elements of order 6. Then either G is a group of period 9, or G is locally finite and one of the following statements holds: Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 16 / 19 New (yet not published) results (E. Jabara, D. V. Lytkina) Suppose G is a group of period 36 which does not possess elements of order 6. Then either G is a group of period 9, or G is locally finite and one of the following statements holds: (1) The Sylow 2-subgroup T of G is nilpotent of class two and normal in G, and the Sylow 3-subgroup is cyclic and acts freely on T ; (2) O2 (G) is a non-trivial elementary abelian group, the Sylow 3-subgroup R of G is cyclic and acts freely on O2 (G). Besides, |G : O2 (G)R| = 2; (3) The Sylow 3-subgroup R of G is abelian and normal in G. The Sylow 2-subgroup T of G acts freely on R and is either a cyclic group, or a quaternion group. Besides, G = RT . Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 16 / 19 (D. V. Lytkina, V. D. Mazurov, A. S. Mamontov). Let G be a group of period 12. If the order of product of any two involutions in G is different from 6 then G is locally finite. This is a generalization of Sanov’s result. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 17 / 19 (D. V. Lytkina, V. D. Mazurov, A. S. Mamontov). Let G be a group of period 12. If the order of product of any two involutions in G is different from 6 then G is locally finite. This is a generalization of Sanov’s result. (V. D. Mazurov, A. S. Mamontov). Let G be a group of period 12. If the order of product of any two involutions in G is different from 4 then G is locally finite. This is a generalization of M. Hall’s result. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 17 / 19 (D. V. Lytkina, V. D. Mazurov, A. S. Mamontov). Let G be a group of period 12. If the order of product of any two involutions in G is different from 6 then G is locally finite. This is a generalization of Sanov’s result. (V. D. Mazurov, A. S. Mamontov). Let G be a group of period 12. If the order of product of any two involutions in G is different from 4 then G is locally finite. This is a generalization of M. Hall’s result. (A. S. Mamontov). Let G be a group of period 12. If G does not contain an element of order 12 then G is locally finite. In other words, a group G with µ(G) = {4, 6} is locally finite. Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 17 / 19 An application Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 18 / 19 An application Problem [G. Cutolo, H. Smith, J. Wiegold. Groups covered by conjugates of proper subgroups. J. Algebra, 293, no.1 (2005), 261-268]. Suppose that the group G is the union of conjugates of a subgroup H. What conditions on H allow us to deduce that G = H? Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 18 / 19 An application Problem [G. Cutolo, H. Smith, J. Wiegold. Groups covered by conjugates of proper subgroups. J. Algebra, 293, no.1 (2005), 261-268]. Suppose that the group G is the union of conjugates of a subgroup H. What conditions on H allow us to deduce that G = H? What about condition that H is finite simple nonabelian? Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 18 / 19 An application Problem [G. Cutolo, H. Smith, J. Wiegold. Groups covered by conjugates of proper subgroups. J. Algebra, 293, no.1 (2005), 261-268]. Suppose that the group G is the union of conjugates of a subgroup H. What conditions on H allow us to deduce that G = H? What about condition that H is finite simple nonabelian? Proposition. Suppose that the group G is the union of conjugates of a subgroup H. If H is isomorphic to one of the following simple groups then H = G: • L2 (Q) where Q is any (locally) finite field of characteristic two; Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 18 / 19 An application Problem [G. Cutolo, H. Smith, J. Wiegold. Groups covered by conjugates of proper subgroups. J. Algebra, 293, no.1 (2005), 261-268]. Suppose that the group G is the union of conjugates of a subgroup H. What conditions on H allow us to deduce that G = H? What about condition that H is finite simple nonabelian? Proposition. Suppose that the group G is the union of conjugates of a subgroup H. If H is isomorphic to one of the following simple groups then H = G: • L2 (Q) where Q is any (locally) finite field of characteristic two; • L2 (9); Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 18 / 19 An application Problem [G. Cutolo, H. Smith, J. Wiegold. Groups covered by conjugates of proper subgroups. J. Algebra, 293, no.1 (2005), 261-268]. Suppose that the group G is the union of conjugates of a subgroup H. What conditions on H allow us to deduce that G = H? What about condition that H is finite simple nonabelian? Proposition. Suppose that the group G is the union of conjugates of a subgroup H. If H is isomorphic to one of the following simple groups then H = G: • L2 (Q) where Q is any (locally) finite field of characteristic two; • L2 (9); • L2 (7). Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 18 / 19 Some open questions Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 19 / 19 Some open questions 1. Let G be a group of exponent 5, 8, 9 or 12. Is G locally finite? Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 19 / 19 Some open questions 1. Let G be a group of exponent 5, 8, 9 or 12. Is G locally finite? 2. Suppose that µ(G) = {4, 5, 6} or {8, 9}. Is G locally finite? Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 19 / 19 Some open questions 1. Let G be a group of exponent 5, 8, 9 or 12. Is G locally finite? 2. Suppose that µ(G) = {4, 5, 6} or {8, 9}. Is G locally finite? 3. Suppose that µ(G) = {5, 6, 11}. Is it true that G is a finite simple group of order 660? Novosibirsk, November 2012 V. D. Mazurov (Novosibirsk) Periodic groups with given element orders 19 / 19

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