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Cyclic groups |
If H
is a subset of a group G, the smallest subgroup of G containing H, denoted
<H> is called the subgroup of G generated by H. Equivalently,
<H> is the intersection of all the subgroups of G which contain H.
If a is an element of G then <a> = {e=a0, a1, a-1,...,an, a-n ...} (where an = a*a*...*a, n times and
a-n = a-1*a-1*...*a-1, n times).
This is called the cyclic subgroup of G generated by a. If all the
integral powers of a are distinct, then <a> is infinite cyclic.
If two powers of a with distinct indices are equal, then there is a least
positive integer n such that an = e and then <a> = {e, a
1,...,an-1}
and we say <a> is cyclic of order n. Two cyclic subgroups are
isomorphic if and only if they have the same order. If <a> has order
n, the subgroups of a are all possible <ad> for all divisors
d of n. The order of <ad> is n/d.
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Permutation groups |
A permutation
of a set A is a 1-1 correspondence f:A -> A. The set of permutations of
A, denoted SymA is a group under the operation of composition.
If |A| = n, we usually regard A as {1,2,...,n} and denote the group Sn,
called the symmetric group of degree n. The order of Sn
is n!
Every finite group of order n is isomorphic
to a subgroup of Sn. Every permutation in Sn can be written as a product of disjoint cycles, and two permutations are conjugate if and only if they have the same cycle structure.
A permutation in Sn is called
a transposition if it switches two elements of {1,2,...,n} (and
leaves the other letters fixed). Every permutation is a product of finitely
many transpositions, because each cycle is. This product is not unique, but its parity
(whether an even or odd number of transpositions) is. A permutation is
even
if it can be written as a product of an even number of transpositions and
Sn if it cannot. Thus the identity map is even. The alternating
group of degree n, An is the subgroup of Sn consisting
of all even permutations. It is normal in Sn.
Sn is generated by a transposition and an n-cycle. To see why this is so, consider the subgroup of Sn generated by (12) and (12...n). Taking the conjugates (12...n)-k(12)(12...n)k = (k,k+1) for k=2,3,...n-1, we get all transpositions whose terms differ by 1.
Example (1234)-1(12)(1234) = (1432)(12)(1234) = (23)
(1234)-2(12)(1234)2=(13)(24)(12)(13)(24) = (34).
Then we have (23)(12)(23) = (13), and taking the conjugates of (13) by powers of (12...n) we get all transpositions whose terms differ by 2.
Example (1234)-1(13)(1234) = (1432)(13)(1234) = (24)
In general, (k,k+1)(1k)(k,k+1) = (1,k+1), and taking conjugates of (1,k+1) by powers of (12...n) we get all transpositions whose terms differ by k. Hence all transpositions are products of (12) and powers of (12...n).
Example (34)(13)(34) = (14), so we have all 6 transpositions in S4.
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Factor groups. |
If H
is a subgroup of a group G, the left coset of H in G by a is aH
= {ah: h in H} and the right coset of H by a is Ha = {ha: h in H}.
They are not usually equal, but they are if and only if H is normal in G. The set of left cosets is a partition of G into disjoint subsets of equal size. So is the set of right cosets. The number of left (or right) cosets is called the index of H in G, denoted [G:H]. Suppose H is normal in G. Then the set G/H of left cosets is also the set of right cosets and G/H is a group under the operation aH*bH = abH , called the factor group of G by H. If G is finite then |G| = |H| |G/H| (Lagrange's Theorem) (If H is not normal, the product of left cosets is not a left coset in general). A subgroup of index 2 (like An in Sn) is always normal. {e} and G are always normal in G. If these are the only normal subgroups of G, we say that G is simple. Last update Dec 14, 1999 Author: Phill Schultz, schultz@maths.uwa.edu.au |
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