Element structure of double cover of alternating group

This article gives specific information, namely, element structure, about a family of groups, namely: double cover of alternating group.
View element structure of group families | View other specific information about double cover of alternating group

Particular cases

$n$	$n!/2$ (order of alternating group $A_{n}$ )	$n!$ (order of the group $2\cdot A_{n}$ )	Alternating group $A_{n}$	The group $2\cdot A_{n}$	Element structure
4	12	24	alternating group:A4	special linear group:SL(2,3)	element structure of special linear group:SL(2,3)
5	60	120	alternating group:A5	special linear group:SL(2,5)	element structure of special linear group:SL(2,5)
6	360	720	alternating group:A6	special linear group:SL(2,9)	element structure of special linear group:SL(2,9)
7	2520	5040	alternating group:A7	double cover of alternating group:A7	element structure of double cover of alternating group:A7
8	20160	40320	alternating group:A8	double cover of alternating group:A8	element structure of double cover of alternating group:A8
9	181440	362880	alternating group:A9	double cover of alternating group:A9	element structure of double cover of alternating group:A9

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 $-1$ .

Splitting criterion

Further information: splitting criterion for conjugacy classes in the alternating group, splitting criterion for conjugacy classes in double cover of alternating group

Consider an unordered integer partition. If the unordered integer partition has an even number of even parts, then it is the cycle type of an even permutation. For even permutations, we would like to know, first, how many conjugacy classes there are in $A_{n}$ with that cycle type. This number is either 1 or 2, depending on whether the conjugacy class splits from $S_{n}$ to $A_{n}$ . For each such conjugacy class, we would like to know whether its inverse image in $2\cdot A_{n}$ splits.

To answer both these questions, we must combined this splitting criterion with the splitting criterion for conjugacy classes in the alternating group. The overall conclusions are below. REMEMBER THAT EVERYTHING BELOW IS FOR PARTITIONS WITH AN EVEN NUMBER OF EVEN PARTS, i.e., those that correspond to even permutations. Note that the first two columns are hypotheses we make about the partition and the third and fourth columns are conclusions we draw about splitting:

Hypothesis: does the partition have at least one even part?	Hypothesis: does the partition have a repeated part? (the repeated part may be even or odd)	Conclusion: does the conjugacy class split from $S_{n}$ to $A_{n}$ in 2?	Conclusion: does the fiber in $2\cdot A_{n}$ over a conjugacy class in $A_{n}$ split in 2?	Total number of conjugacy classes in $2\cdot A_{n}$ corresponding to this partition (4 if Yes to both preceding columns, 2 if Yes to one and No to other, 1 if No to both)
No	No	Yes	Yes	4
No	Yes	No	Yes	2
Yes	No	No	Yes	2
Yes	Yes	No	No	1

Combinatorics

Partitions that correspond to four conjugacy classes

We have canonical bijections:

Conjugacy classes from $S_{n}$ that split in $A_{n}$ and then again in $2\cdot A_{n}$ $\leftrightarrow$ Conjugacy classes from $S_{n}$ that split in $A_{n}$ $\leftrightarrow$ Partitions of $n$ into distinct odd parts $\leftrightarrow$ 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 use the letter $A$ to denote the number of such partitions of $n$ .

Partitions that split in alternating group but not in its double cover

By the table above, there are two such types of partitions:

Partitions with no even parts but with a repeated odd part
Partitions with a positive even number of even parts but with no repeated parts (even or odd)

For this page, we will use the letter $P$ and $Q$ to denote the numbers of such partitions of $n$ .

Partitions that don't split either in alternating group or in its double cover

By the table above, these are the partitions with a positive even number of even parts (there may also be odd parts), and with at least one repeated part (even or odd). For this page, we will use the letter $U$ to denote the number of such partitions of $n$ .

Totals

We have the following about the total number of conjugacy classes:

Number of conjugacy classes of $S_{n}$ = Number of unordered integer partitions of $n$ = $A+2P+2Q+2U$
Number of conjugacy classes of $A_{n}$ = $2A+P+Q+U$
Number of conjugacy classes of $2\cdot A_{n}$ = $4A+2P+2Q+U$

We thus have the formula:

Number of conjugacy classes of $2\cdot A_{n}$ = Number of conjugacy classes of $S_{n}$ + $3A-U$

Or more explicitly:

Number of conjugacy classes of $2\cdot A_{n}$ = Number of unordered integer partitions of $n$ + 3(Number of partitions of $n$ into distinct odd parts) - (Number of partitions of $n$ with a positive even number of even parts and possibly also odd parts, and with at least one repeated part)

Orders of elements

For a given unordered integer partition, the lcm of the parts gives the order of any element of $S_{n}$ with that partition as its cycle type. If the partition has an even number of even parts, the conjugacy class is in $A_{n}$ and the order as an element of $A_{n}$ is the same: the lcm of the parts.

When we take the inverse image in $2\cdot A_{n}$ , the order of elements in the inverse image is either the same or twice the order of the element in $A_{n}$ . We provide below a slight modification of the previous table that includes order information:

Hypothesis: does the partition have at least one even part?	Hypothesis: does the partition have a repeated part? (the repeated part may be even or odd)	Conclusion: does the conjugacy class split from $S_{n}$ to $A_{n}$ in 2?	Conclusion: does the fiber in $2\cdot A_{n}$ over a conjugacy class in $A_{n}$ split in 2?	Total number of conjugacy classes in $2\cdot A_{n}$ corresponding to this partition (4 if Yes to both preceding columns, 2 if Yes to one and No to other, 1 if No to both)	Number of these conjugacy classes where order of element = lcm of parts	Number of these conjugacy classes where order of element = twice the lcm of parts
No	No	Yes	Yes	4	2	2
No	Yes	No	Yes	2	1	1
Yes	No	No	Yes	2	0	2
Yes	Yes	No	No	1	0	1

Number of equivalence classes of various kinds

$n$	Double cover of alternating group $2\cdot A_{n}$	Order equals $n!$	$A=$ number of partitions of $n$ into distinct odd parts	$P=$ number of partitions of $n$ into odd parts with at least one repetition	$Q=$ number of partitions of $n$ into distinct parts with a positive even number of even parts	$U=$ number of partitions of $n$ with a positive even number of even parts and at least one repeated part (even or odd)	Number of conjugacy classes in $S_{n}$ equals $A+2P+2Q+2U$	Number of conjugacy classes in $A_{n}$ equals $2A+P+Q+U$	Number of conjugacy classes in $2\cdot A_{n}$ equals $4A+2P+2Q+U$
4	special linear group:SL(2,3)	24	1	1	0	1	5	4	7
5	special linear group:SL(2,5)	120	1	2	0	1	7	5	9
6	special linear group:SL(2,9)	720	1	3	1	1	11	7	13
7	double cover of alternating group:A7	5040	1	4	1	2	15	9	16
8	double cover of alternating group:A8	40320	2	4	1	5	22	14	23