Download MS3001 — Exercise Sheet 4

24/11/03
MS3001 — Exercise Sheet 4
1. (a) Consider the following binary operations on Z. In each case decide if the operation
is commutative, associative, has an identity, and, if it has an identity, which elements
have inverses.
(i)∗ a ∗1 b = a + b − ab
(iii)∗ a ∗3 b = min{a, b}
(ii) a ∗2 b = 2a2 + b2
(iv) a ∗4 b = b
(b) Consider the binary operation on N defined by a ∗ b = ab . Show that ∗ is not
associative, is not commutative and does not have an inverse.
(c)∗ Let 5Z = {. . . , −10, −5, 0, 5, 10, . . .}. Show that the binary operation of addition on
5Z makes this set into a group.
2.∗ Is G = (Z, −) a group?
3.∗ Show that the operation ∗ defined on GL(2, R) below is commutative, has an identity,
and that every element in GL(2, R) has an inverse, but that ∗ is not associative.
S ∗ T = ST + T S.
4. For any nonempty set A let P (A) denote the set of all subsets of A. Consider the following
binary operations on P (A): (i) (P (A), ∩)
(ii)∗ (P (A), −)
(iii) (P (A), 4).
Show that the operation in (i) is commutative, associative and has an identity, but that
only one set from P (A) has an inverse.
Show that the operation in (ii) is not commutative, not associative and does not have an
identity.
Show that the operation in (iii) turns P (A) into an abelian group.
5. Find an example of a binary operation on a finite set that is not associative. Can you
find one for which the Cayley table is a Latin square, that is, for which each element of
the set appears exactly once in each row and in each column?
6. Given the following sets and associative operations, write out the relevant Cayley table
and decide if it is a group. If so, list the inverses of the elements:
(i)∗ {(1), (134), (143)} ⊂ S4 with the usual product on permutations.
(ii) {(1), (12), (134), (234)} ⊂ S4 with the usual product on permutations.
(iii) {0, 2, 4, 6} with addition modulo 8.
7. Define G to be the set
a b
: a, b, c, d ∈ R, ad − bc = 1
c d
equipped with the usual matrix multiplication. Show that G is a nonabelian group. (You
may assume associativity of matrix multiplication. Can you explain why?)
[This group is usually denoted by SL(2, R).]
8. Show that the following sets of 2 × 2 matrices form groups under the usual matrix multiplication. Are either of these groups abelian?
a 0
a b
∗
(i)
: a, d ∈ R, ad 6= 0
(ii)
: a, b, d ∈ R, ad 6= 0
0 d
0 d
9.∗ If G is an abelian group, show that (gh)2 = g 2 h2 for all g, h ∈ G.
Conversely, suppose that G is a group for which (gh)2 = g 2 h2 holds for all g, h ∈ G. Show
that G is necessarily abelian.
Give an example of elements α, β in S3 that do not satisfy (αβ)2 = α2 β 2 .
10. Let G be a group in which every element is its own inverse. Show that G is abelian.
11. Show that the following subsets of the rational numbers Q are both subgroups of the
group (Q∗ , ×) where, as usual, Q∗ = Q − {0}:
1 + 2m
∗
(i) {x ∈ Q : x > 0}
(ii)
: m, n ∈ Z
1 + 2n
12. Consider the following groups:
(i) G = ({1, 3, 5, 7}, ×8 )
(ii) H = ({1, 2, 4, 5, 7, 8}, ×9 )
Show that G is not cyclic, but that H is cyclic. List all of the generators of H.
13. Let G and H be groups. Define a binary operation on the Cartesian product G × H by
(g1 , h1 )(g2 , h2 ) = (g1 g2 , h1 h2 ).
Show that this operation makes G × H into a group. Furthermore show that G × H is an
abelian group if and only if G and H are both abelian.
Consider the cyclic groups (Z2 , +2 ) and (Z3 , +3 ). Write out the Cayley tables for Z2 × Z3
and Z2 × Z2 and hence show that the former is cyclic, but that the latter is not cyclic.
14.∗ Let G be a group and let g, h ∈ G. Show that gh and hg have the same order.
[Hint: prove by induction that h(gh)n g = (hg)n+1 for all n ≥ 1. Use this to show that
(gh)n = e if and only if (hg)n = e.]
15. (a) Let G be a group, H a subgroup of G and g ∈ G. Show that H = Hg if and only
if g ∈ H. More generally, for any two elements g1 , g2 ∈ G, show that Hg1 = Hg2 if and
only if g1 g2−1 ∈ H.
(b) Find a subgroup H of S3 and a permutation α ∈ S3 such that αH 6= Hα, where
αH = {αβ : β ∈ H} and Hα = {βα : β ∈ H}.
16. Find all of the subgroups of (Z9 , +9 ).
17. The following are parts of the Cayley tables for two groups on sets G = {a, b, c, d} and
H = {v, w, x, y, z}. Show that there is only one way to fill each of them in.
a
b
c
d
(i)
∗
a
c
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b
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c
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d
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a
d
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(ii)∗
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