Overview / review of basic group theory

For: Math 420, 421 at Northern Illinois University
From: Beachy/Blair, Abstract Algebra, Second Edition
Covering: Sections 3.1 through 3.6

Groups, in general

3.1.3. Definition. A group (G, * ) is a nonempty set G together with a binary operation * on G such that the following conditions hold:
(i) Closure: For all a,b in G the element a * b is a uniquely defined element of G.
(ii) Associativity: For all a,b,c in G , we have a * (b * c) = (a * b) * c .
(iii) Identity: There exists an identity element e in G such that e * a = a and a * e = a , for all a in G.
(iv) Inverses: For each a in G there exists an inverse element b in G such that a * b = e and b * a = e .

3.1.6. Proposition. (Cancellation Property for Groups) Let G be a group, and let a,b,c be elements of G.
(a) If ab = ac , then b = c .
(b) If ac = bc , then a = b .

3.1.8. Definition. A group G is said to be abelian if a b = b a for all a,b in G.

3.1.9. Definition. A group G is said to be a finite group if the set G has a finite number of elements. In this case, the number of elements is called the order of G, denoted by | G |.

3.2.7. Definition. Let a be an element of the group G. If there exists a positive integer n such that an = e, then a is said to have finite order, and the smallest such positive integer is called the order of a , denoted by o(a) .
If there does not exist a positive integer n such that an = e, then a is said to have infinite order.

3.2.1. Definition. Let G be a group, and let H be a subset of G. Then H is called a subgroup of G if H is itself a group, under the operation induced by G .

3.2.2. Proposition. Let G be a group with identity element e , and let H be a subset of G. Then H is a subgroup of G if and only if the following conditions hold:
(i) ab is in H for all a,b in H;
(ii) e is in H;
(iii) for all a in H, the inverse of a is also in H.

3.2.10. Theorem. (Lagrange) If H is a subgroup of the finite group G, then the order of H is a divisor of the order of G.

3.2.11. Corollary. Let G be a finite group of order n .
(a) For any a in G, o(a) is a divisor of n .
(b) For any a in G, an = e.

3.2.12. Corollary. Any group of prime order is cyclic.

3.4.1. Definition. Let G and G' be groups, and let f : G -> G' be a function. Then f is said to be a group isomorphism if f is one-to-one and onto and
f (ab) = f (a) f (b)
for all a,b in G. In this case, G is said to be isomorphic to G'.

3.4.3. Proposition. Let f : G -> G' be an isomorphism of groups.
(a) If a has order n in G, then f (a) has order n in G'.
(b) If G is abelian, then so is G'.
(c) If G is cyclic, then so is G'.

Cyclic groups

3.2.5 Definition. Let G be a group, and let a be any element of G. The set
< a > = { x in G | x = an for some n in Z }
is called the cyclic subgroup generated by a .
The group G is called a cyclic group if there exists an element a in G such that G = < a > . In this case a is called a generator of G.

3.2.6 Proposition. Let G be a group, and let a be in G.
(a) The set < a > is a subgroup of G.
(b) If K is any subgroup of G such that a in K, then < a > is a subset of K.

3.2.8. Proposition. Let a be an element of the group G.
(a) If a has infinite order, and ak = am for integers k , m , them k = m .
(b) If a has finite order and k is any integer, then ak = e if and only if o(a) | k .
(c) If a has finite order o(a) = n , then for all integers k,m , we have ak = am if and only if k is congruent to m modulo n . Furthermore, | < a > | = o(a) .

Corollaries to Lagrange's Theorem (restated):
(a) For any a in G, o(a) is a divisor of | G |.
(b) For any a in G, an = e, for n = | G |.
(c) Any group of prime order is cyclic.

3.5.1. Theorem. Every subgroup of a cyclic group is cyclic.

3.5.2 Theorem. Let G be a cyclic group.
(a) If G is infinite, then G is isomorphic to the group Z of integers.
(b) If | G | = n , then G is isomorphic to the group Zn of integers modulo n .

3.5.3. Proposition. Let G = < a > be a cyclic group with order n.
(a) If m is any integer, then < am > = < ad > , where d = gcd (m,n), and am has order n / d .
(b) The element ak generates G if and only if gcd (k,n) = 1.
(c) The subgroups of G are in one-to-one correspondence with the positive divisors of n .
(d) If m and k are divisors of n , then < am > is a subset of < ak > if and only if k | m .

Permutation groups

3.1.4. Definition. The set of all permutations of a set S is denoted by Sym(S).
The set of all permutations of the set { 1,2,...,n } is denoted by Sn.

3.1.5. Proposition. If S is any nonempty set, then Sym(S) is a group under the operation of composition of functions.

2.3.5. Theorem. Every permutation in Sn can be written as a product of disjoint cycles. The cycles that appear in the product are unique.

2.3.8 Proposition. If a permutation in Sn is written as a product of disjoint cycles, then its order is the least common multiple of the lengths of its cycles.

3.6.1. Definition. Any subgroup of the symmetric group Sym(S) on a set S is called a permutation group.

3.6.2. Theorem. (Cayley) Every group is isomorphic to a permutation group.

3.6.3. Definition. Let n > 2 be an integer. The group of rigid motions of a regular n-gon is called the nth dihedral group, denoted by Dn.

We can describe Dn as { ak, ak b | 0 <= k < n }, subject to the relations o(a) = n , o(b) = 2 , and b a = a-1b.

2.3.11. Theorem. If a permutation is written as a product of transpositions in two ways, then the number of transpositions is either even in both cases or odd in both cases.

2.3.12. Definition. A permutation is called even if it can be written as a product of an even number of transpositions, and odd if it can be written as a product of an odd number of transpositions.

3.6.4. Proposition. The set of all even permutations of Sn is a subgroup of Sn.

3.6.5. Definition. The set of all even permutations of Sn is called the alternating group on n elements, and will be denoted by An.

Other examples

Example 3.1.4. (Group of Units Modulo n ) Let n be a positive integer. The set of units modulo n is an abelian group under multiplication of congruence classes. Its order is given by the value at n of the Euler phi-function.

3.1.10. Definition. The set of all invertible n x n matrices with entries in R is called the general linear group of degree n over the real numbers, and is denoted by GLn (R).

3.1.11. Proposition. The set GLn (R) forms a group under matrix multiplication.

3.3.3. Definition. Let G and G' be groups. The set of all ordered pairs ( a , a' ) such that a is in G and a' is in G' is called the direct product of G and G' , denoted by
G x G' .

3.3.4. Proposition. Let G and G' be groups.
(a) The direct product G x G' of G and G' is a group under the multiplication
( a , a' ) ( b , b' ) = ( a b , a' b' ) .
(b) If the elements a in G and a' in G' have orders n and m , respectively, then in
G x G' the element ( a , a' ) has order lcm [n,m].

3.4.5. Proposition. If m,n are positive integers such that gcd (m,n) = 1 , then the direct product
(Zm) x (Zn) is isomorphic to Zmn.


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