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'.
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 .
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.
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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