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Transcript
Groups
Definition: A group is a non-empty set G equipped with a binary operation * that
satisfies the following axioms:
1. Closure: If a ∈ G and b ∈ G , then a ∗ b ∈ G .
2. Associativity: a ∗ ( b ∗ c ) = ( a ∗ b ) ∗ c for all a, b, c ∈ G .
3. There is an element e ∈ G (called the identity) such that a ∗ e = e ∗ a = a for every
a ∈ G.
4. For each a ∈ G , there is an element d ∈ G (called the inverse of a) such that
a ∗ d = e = d ∗ a.
A group is said to be abelian if it also satisfies
5. Commutativity: a ∗ b = b ∗ a for all a, b ∈ G .
A group is said to be finite (or of finite order) if it has a finite number of elements. In this
case the number of elements in G is called the order of G and is denoted by o(G) or |G|.
A group with infinitely many elements is said to have infinite order.
We have considered the following examples of finite groups:
1. Dn , (the dihedral groups of order 2n) corresponding to the groups of symmetries
of the regular n-gons.
2. Sn , (the permutation groups of order n!) corresponding to the group of bijections
of the set {1, 2, 3, …, n} to itself.
3. Zn , (the set of integers modulo n) composed of the equivalence classes of integers
modulo n, under the operation of addition.
Problem 1: Give an example of an infinite abelian group as well as an infinite nonabelian group.
Theorem: Let G and H be groups. Define G × H to be the set of ordered pairs (g,h)
where g ∈ G and h ∈ H . Define an operation • on G × H by
( g, h ) • ( g′, h ′) = ( g ∗ g′, h ∗ h ′) .
Problem 2:
(a) Show that G × H is a group.
(b) Show that if G and H are both abelian, then so is G × H .
Problem 3: Give examples of nonabelian groups of orders 12, 16, 30, and 48.
Problem 4: If G is a group under the stated operation, prove it; if not, give a
conterexample:
(a) G is the set of rational numbers Q; a ∗ b = a + b + 3
ab
(b) G is the set of non-zero rational numbers; a ∗ b =
3
(c) G=GL(2,R), the set of 2 × 2 invertible matrices with real entries; the operation is the
usual matrix product.
We will prove the following in class:
Theorem: Let G be a group and let a, b, c ∈ G . Then
(1) G has a unique identity element.
(2) Cancellation holds in G:
If a ∗ b = a ∗ c , then b = c . If b ∗ a = c ∗ a , then b = c .
(3) Each element of G has a unique inverse.
Since inverses are unique, we can denote the inverse of a by a−1 .
Problem 5: Let G be a group and let a, b, c ∈ G . Prove
(1) ( ab )−1 = b−1a−1.
(2) ( a−1 )−1 = a .
Definition: An element a in a group is said to have finite order if a k = e for some
positive integer k. In this case, the order of the element a is the smallest positive integer
n for which a n = e . The order of a is denoted by o(a) or |a|. An element a is said to have
infinite order if a k ≠ e for every positive integer k.
Examples:
(a) In the group of nonzero rationals under multiplication, every element other than 1 and
–1 has infinite order.
(b) In the group of integers modulo n under addition, every element has finite order.
Problem 6:
(a) GL(2,R) is the set of 2 × 2 invertible matrices with real entries, as in problem 4(c).
0 1
0 −1
 has order 3 and B = 
 has order 4.
Show that A = 
−1 −1
1 0 
(b) Show that AB has infinite order.
We will prove the following in class:
Theorem: Let G be a group and a ∈ G .
(1) If a has infinite order, then the elements a k are distinct (k an integer.)
(2) If a has finite order n, then a k = e if and only if n|k , and a i = a j if and only if
i ≡ j (mod n).
(3) If a has order n and n=td with d>0, then a t has order d.
Problem 7: Let G be a group and let a ∈ G . If a i = a j with i ≠ j , then a has finite order.
Problem 8: True or false: A group of order n contains and element of order n. Justify
your answer.