Element structure of alternating group:A5: Difference between revisions

Revision as of 00:56, 30 October 2010

This article gives specific information, namely, element structure, about a particular group, namely: alternating group:A5.
View element structure of particular groups | View other specific information about alternating group:A5

This article gives the element structure of alternating group:A5.

Conjugacy class structure

There is a total of 5 conjugacy classes, of which 3 are unsplit from symmetric group:S5, and 2 are a split pair arising from a single conjugacy class in $S_{5}$ . The conjugacy class sizes are 1, 20, 15, 12, 12.

Interpretation as alternating group

For a symmetric group, cycle type determines conjugacy class. The statement is almost true for the alternating group, except for the fact that some conjugacy classes of even permutations in the symmetric group split into two in the alternating group, as per the splitting criterion for conjugacy classes in the alternating group, which says that a conjugacy class of even permutations splits in the alternating group if and only if it is the product of odd cycles of distinct length.

Here are the unsplit conjugacy classes:

Partition	Verbal description of cycle type	Representative element of the cycle type	All elements of the cycle type	Size of conjugacy class	Formula for size	Element order
1 + 1 + 1 + 1 + 1	five fixed points	$()$ -- the identity element	$()$	1	$\frac{5!}{(1)^{5} (5!)}$	1
3 + 1 + 1	one 3-cycle, two fixed points	$(1, 2, 3)$	[SHOW MORE] $(1, 2, 3)$ , $(1, 3, 2)$ , $(1, 2, 4)$ , $(1, 4, 2)$ , $(1, 2, 5)$ , $(1, 5, 2)$ , $(1, 3, 4)$ , $(1, 4, 3)$ , $(1, 3, 5)$ , $(1, 5, 3)$ , $(1, 4, 5)$ , $(1, 5, 4)$ , $(2, 3, 4)$ , $(2, 4, 3)$ , $(2, 3, 5)$ , $(2, 5, 3)$ , $(2, 4, 5)$ , $(2, 5, 4)$ , $(3, 4, 5)$ , $(3, 5, 4)$	20	$\frac{5!}{(3) (1)^{2} (2!)}$	3
2 + 2 + 1	double transposition: two 2-cycles, one fixed point	$(1, 2) (3, 4)$	[SHOW MORE] $(1, 2) (3, 4)$ , $(1, 3) (2, 4)$ , $(1, 4) (2, 3)$ , $(2, 3) (4, 5)$ , $(2, 4) (3, 5)$ , $(2, 5) (3, 4)$ , $(1, 3) (4, 5)$ , $(1, 4) (3, 5)$ , $(1, 5) (3, 4)$ , $(1, 2) (4, 5)$ , $(1, 4) (2, 5)$ , $(1, 5) (2, 4)$ , $(1, 2) (3, 5)$ , $(1, 3) (2, 5)$ , $(1, 5) (2, 3)$	15	$\frac{5!}{(2)^{2} (2!) (1)}$	2

Here is the split pair of conjugacy classes:

Partition	Verbal description of cycle type	Combined size of conjugacy classes	Formula for combined size	Size of each half	Representative of first half	Representative of second half	Real?	Rational?	Element order
5	one 5-cycle	24	$\frac{5!}{5}$	12	$(1, 2, 3, 4, 5)$	$(1, 3, 5, 2, 4)$	Yes	No	5

Interpretation as projective special linear group of degree two over field:F5

Compare with element structure of projective special linear group of degree two#Conjugacy class structure.

Nature of conjugacy class upstairs in $S L_{2}$	Eigenvalues	Characteristic polynomial	Minimal polynomial	Size of conjugacy class	Number of such conjugacy classes	Total number of elements	Representative matrices upstairs in $S L_{2}$ (one per conjugacy class)	Representative element as permutation
Diagonalizable over field:F5 with equal diagonal entries, hence a scalar	${1, 1}$ or ${- 1, - 1}$ , both correspond to the same element	$(x - a)^{2}$ where $a \in {- 1, 1}$	$x - a$ where $a \in {- 1, 1}$	1	1	1	$(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$	$()$
Not diagonal, has Jordan block of size two	$1$ (multiplicity 2) or $- 1$ (multiplicity 2). Each conjugacy class has one representative of each type.	$(x - a)^{2}$ where $a \in {- 1, 1}$	Same as characteristic polynomial	12	2	24	$(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$ , $(\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix})$	$(1, 2, 3, 4, 5)$ , $(1, 3, 5, 2, 4)$
Diagonlizable over field:F5 with diagonal entries multiplying to $- 1$	${2, 3}$	$x^{2} + 1$	$x^{2} + 1$	15	1	15	$(\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix})$	$(1, 2) (3, 4)$
Diagonalizable over field:F25, not over field:F5. Must necessarily have no repeated eigenvalues.	Pair of conjugate elements of field:F25 of norm 1. Each pair identified with its negative pair.	$x^{2} - x + 1$ , $x^{2} + x + 1$ , get identified.	Same as characteristic polynomial	20	1	20	$(\begin{matrix} 0 & - 1 \\ 1 & - 1 \end{matrix})$	$(1, 2, 3)$
Diagonalizable over field:F5 with distinct (and hence mutually inverse) diagonal entries, whose product is not $- 1$	None	--	--	--	0	0	--	--
Total	NA	NA	NA	NA	5	60	NA	NA

Interpretation as special linear group of degree two over field:F4

Compare with element structure of special linear group of degree two#Conjugacy class structure.

Nature of conjugacy class	Eigenvalues	Characteristic polynomial	Minimal polynomial	Size of conjugacy class	Number of such conjugacy classes	Total number of elements	Semisimple?	Diagonalizable over $F_{q}$ ?	Splits in $S L_{2}$ relative to $G L_{2}$ ?	Representative matrices (one for each conjugacy class)	Representative element as permutation
Diagonalizable over field:F4 with distinct (and hence mutually inverse) diagonal entries	$λ, 1 / λ$ where $λ \in F_{4} ∖ {0, 1}$	$x^{2} + x + 1$	$x^{2} + x + 1$	20	1	20	Yes	Yes	No	$(\begin{matrix} 0 & 1 \\ 1 & 1 \end{matrix})$	$(1, 2, 3)$
Diagonalizable over field:F16, not over field:F4. Must necessarily have no repeated eigenvalues.	Pair of conjugate elements of field:F16 of norm 1	$x^{2} - a x + 1$ , $a \neq 0, 1$ .	Same as characteristic polynomial	12	2	24	Yes	No	No	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	$(1, 2, 3, 4, 5)$ , $(1, 3, 5, 2, 4)$
Diagonalizable over field:F4 with equal diagonal entries, hence a scalar.	$1, 1$	$x^{2} + 1$	$x + 1$	1	1	1	Yes	Yes	No	$(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$	$()$
Not diagonal, has Jordan block of size two	$1$ (multiplicity 2)	$x^{2} + 1$	$x^{2} + 1$	15	1	15	No	No	No	$(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$	$(1, 2) (3, 4)$
Total	NA	NA	NA	NA	5	60	45			NA	NA

@@ Line 43: / Line 43: @@
 ! Nature of conjugacy class upstairs in <math>SL_2</math> !! Eigenvalues !! Characteristic polynomial !! Minimal polynomial !! Size of conjugacy class!! Number of such conjugacy classes !! Total number of elements !! Representative matrices upstairs in <math>SL_2</math> (one per conjugacy class) !! Representative element as permutation
 |-
-| Diagonalizable over [[field:F5]] with ''distinct'' (and hence mutually inverse) diagonal entries, whose product is not <math>-1</math> || None || -- || -- ||  || 0 || 0 || -- || --
+| Diagonalizable over [[field:F5]] with equal diagonal entries, hence a scalar || <math>\{ 1,1 \}</math> or <math>\{ -1,-1\}</math>, both correspond to the same element || <math>(x - a)^2</math> where <math>a \in \{ -1,1 \}</math> || <math>x - a</math> where <math>a \in \{ -1,1\}</math> || 1 || 1 || 1 || <math>\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}</math> || <math>()</math>
+|-
+| Not diagonal, has Jordan block of size two  || <math>1</math> (multiplicity 2) or <math>-1</math> (multiplicity 2). Each conjugacy class has one representative of each type. || <math>(x - a)^2</math> where <math>a \in \{ -1,1 \}</math> || Same as characteristic polynomial|| 12 || 2 || 24 || <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & 2 \\ 0 & 1 \\\end{pmatrix}</math> || <math>(1,2,3,4,5)</math>, <math>(1,3,5,2,4)</math>
 |-
 | Diagonlizable over [[field:F5]] with diagonal entries multiplying to <math>-1</math> || <math>\{ 2,3 \}</math> || <math>x^2 + 1</math> || <math>x^2 + 1</math> || 15 || 1 || 15 || <math>\begin{pmatrix} 2 & 0 \\ 0 & 3 \\\end{pmatrix}</math> || <math>(1,2)(3,4)</math>
@@ Line 49: / Line 51: @@
 | Diagonalizable over [[field:F25]], not over [[field:F5]]. Must necessarily have no repeated eigenvalues. || Pair of conjugate elements of [[field:F25]] of norm 1. Each pair identified with its negative pair. || <math>x^2 - x + 1</math>, <math>x^2 + x + 1</math>, get identified. || Same as characteristic polynomial || 20 || 1 || 20 || <math>\begin{pmatrix} 0 & -1 \\ 1 & -1 \\\end{pmatrix}</math> || <math>(1,2,3)</math>
 |-
-| Diagonalizable over [[field:F5]] with equal diagonal entries, hence a scalar || <math>\{ 1,1 \}</math> or <math>\{ -1,-1\}</math>, both correspond to the same element || <math>(x - a)^2</math> where <math>a \in \{ -1,1 \}</math> || <math>x - a</math> where <math>a \in \{ -1,1\}</math> || 1 || 1 || 1 || <math>\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}</math> || <math>()</math>
+| Diagonalizable over [[field:F5]] with ''distinct'' (and hence mutually inverse) diagonal entries, whose product is not <math>-1</math> || None || -- || -- || -- || 0 || 0 || -- || --
-|-
-| Not diagonal, has Jordan block of size two  || <math>1</math> (multiplicity 2) or <math>-1</math> (multiplicity 2). Each conjugacy class has one representative of each type. || <math>(x - a)^2</math> where <math>a \in \{ -1,1 \}</math> || Same as characteristic polynomial|| 12 || 2 || 24 || <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & 2 \\ 0 & 1 \\\end{pmatrix}</math> || <math>(1,2,3,4,5)</math>, <math>(1,3,5,2,4)</math>
 |-
 | Total || NA || NA || NA || NA || 5 || 60 || NA || NA