Element structure of alternating groups

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This article discusses the element structure of the alternating group A_n of finite degree n. Note that for n = 0,1, the alternating group coincides with the symmetric group, and for n = 2, it is trivial, so the interesting behavior begins from n = 3.

The article builds heavily on the element structure of symmetric groups. In particular, we use the fact that cycle type determines conjugacy class in the symmetric group S_n. A_n is viewed naturally as the subgroup of S_n comprising even permutations.

Particular cases

n Alternating group Order Element structure page
3 cyclic group:Z3 3 element structure of cyclic group:Z3
4 alternating group:A4 12 element structure of alternating group:A4
5 alternating group:A5 60 element structure of alternating group:A5
6 alternating group:A6 360 element structure of alternating group:A6
7 alternating group:A7 2520 element structure of alternating group:A7
8 alternating group:A8 20160 element structure of alternating group:A8
9 alternating group:A9 181440 element structure of alternating group:A9
10 alternating group:A10 1814400 element structure of alternating group:A10

Basic facts

Condition for a permutation to be even

Further information: even permutation

A permutation is an even permutation, i.e., a member of the alternating group, if and only if the number of cycles of even length (which are thus odd cycles) in its cycle decomposition is even. Thus, we can count the number of occurrences of even numbers in the cycle type of a permutation to determine whether the permutation is even. If a permutation is not even, it is odd.

Note that any conjugacy class in S_n is either contained completely in A_n (which means all the permutations are even) or it is contained completely outside A_n (which means all the permutations are odd). This is another way of saying that A_n is a normal subgroup of S_n; it is in fact a subgroup of index two and index two implies normal. It is also the kernel of the sign homomorphism that sends even permutations to +1 and odd permutations to -1.

Condition for a conjugacy class to split from S_n to A_n

Further information: splitting criterion for conjugacy classes in the alternating group

For a conjugacy class of even permutations in S_n, the conjugacy class is contained inside A_n. There are two possibilities:

  1. It remains a single conjugacy class inside A_n (the unsplit case): This happens if it is centralized by some odd permutation. In terms of cycle type, this is equivalent to saying that it either contains a cycle of even length or it contains two cycles of equal odd length.
  2. It splits into two conjugacy classes inside A_n (the split case): This happens if its centralizer in S_n is completely contained in A_n. In terms of cycle type, this is equivalent to saying that all its cycles have distinct odd length.

Note that when a conjugacy class splits, the two split halves are still automorphic to each other in A_n. The conjugation by an odd permutation in S_n restricts to an outer automorphism on the normal subgroup A_n.

Condition for a split conjugacy class in A_n to be real

Further information: criterion for element of alternating group to be real

Any element whose conjugacy class is unsplit from S_n, is a real element, and in fact a rational element of A_n.

For an element whose conjugacy class is split from S_n, the criterion to determine whether it is real or not is as follows: if r_1, r_2, \dots, r_k are the distinct odd cycle lengths of the permutation, it is real if and only if \sum_{i=1}^k (r_i - 1)/2 is even. In other words, the number of r_is that are congruent to 3 modulo 4 should be even.

Combinatorics

Split conjugacy classes

We have canonical bijections:

Conjugacy classes from S_n that split in A_n \leftrightarrow Partitions of n into distinct odd parts (via splitting criterion) \leftrightarrow Self-conjugate unordered integer partitions of n \leftrightarrow Irreducible representations of S_n that split in A_n

For more on these bijections, see set of self-conjugate unordered integer partitions

For this page, we will denote this number by A.

Non-split conjugacy classes

We have equalities:

Number of conjugacy classes of even permutations in S_n that don't split in S_n = Number of conjugacy classes of odd permutations in S_n = Number of (conjugate) pairs of non-self-conjugate partitions of n = Number of pairs (in the sense of restricting to the same thing on A_n) of irreducible representations of S_n that don't split in A_n.

For this page, we will denote this number by B.

Totals

If A equals the number of conjugacy classes from S_n that split in A_n, and B equals the number that don't, then:

  • The number of conjugacy classes in A_n is 2A + B
  • The number of conjugacy classes in S_n is A + 2B

Numerical information in particular cases

Number of equivalence classes of various kinds

n Alternating group A_n Order A= number of conjugacy classes from S_n that split in A_n B= number of conjugacy classes from S_n that don't split in A_n 2A + B = number of conjugacy classes in A_n A + 2B = number of conjugacy classes in S_n C = number of conjugacy classes from S_n which, even after splitting, remain real 2C + B = number of conjugacy classes of real elements in A_n A + B + C= number of equivalence classes under real conjugacy in A_n
3 cyclic group:Z3 3 1 1 3 3 0 1 2
4 alternating group:A4 12 1 2 4 5 0 2 3
5 alternating group:A5 60 1 3 5 7 1 5 5
6 alternating group:A6 360 1 5 7 11 1 7 7
7 alternating group:A7 2520 1 7 9 15 0 7 8
8 alternating group:A8 20160 2 10 14 22 0 10 12
9 alternating group:A9 181440 2 14 18 30 1 16 17
10 alternating group:A10 1814400 2 20 24 42 2 24 24