Quiz:Special linear group:SL(2,3)

For background, see special linear group:SL(2,3).

Subgroups

See subgroup structure of special linear group:SL(2,3) for more information.

Basic stuff

$SL(2,3)$ has order 24. Summary table on the structure of subgroups: [SHOW MORE]

Quick summary

Item	Value
number of subgroups	15
number of conjugacy classes of subgroups	7
number of automorphism classes of subgroups	7
isomorphism classes of Sylow subgroups, corresponding fusion systems, and Sylow numbers	2-Sylow: quaternion group (order 8) as Q8 in SL(2,3) with its non-inner fusion system (see non-inner fusion system for quaternion group), Sylow number 1 3-Sylow: cyclic group:Z3 with its non-inner fusion system, Sylow number 4
Hall subgroups	The order has only two prime divisors, so no possibility of Hall subgroups other than trivial subgroup, whole group, and Sylow subgroups
maximal subgroups	There are maximal subgroups of orders 6 (Z6 in SL(2,3)) and 8 (2-Sylow subgroup of special linear group:SL(2,3))
normal subgroups	There are two proper nontrivial normal subgroups: center of special linear group:SL(2,3) and 2-Sylow subgroup of special linear group:SL(2,3)

Table classifying subgroups up to automorphism

Note that, in the matrices, -1 can be written as 2 since elements are taken modulo 3.

Automorphism class of subgroups	Representative subgroup	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (=1 iff automorph-conjugate subgroup)	Size of each conjugacy class (=1 iff normal subgroup)	Total number of subgroups (=1 iff characteristic subgroup)	Isomorphism class of quotient (if exists)	Subnormal depth (if subnormal)	Note
trivial subgroup	$\{{\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}\}$	trivial group	1	24	1	1	1	special linear group:SL(2,3)	1	trivial
center of special linear group:SL(2,3)	$\{{\begin{pmatrix}1&0\\0&1\\\end{pmatrix}},{\begin{pmatrix}-1&0\\0&-1\\\end{pmatrix}}\}$	cyclic group:Z2	2	12	1	1	1	alternating group:A4	1
Z4 in SL(2,3)	$\langle {\begin{pmatrix}0&1\\-1&0\\\end{pmatrix}}\rangle$	cyclic group:Z4	4	6	1	3	3	--	2
2-Sylow subgroup of special linear group:SL(2,3)	$\langle {\begin{pmatrix}0&1\\-1&0\\\end{pmatrix}},{\begin{pmatrix}-1&-1\\-1&1\\\end{pmatrix}}\rangle$	quaternion group	8	3	1	1	1	cyclic group:Z3	1	2-Sylow
Z3 in SL(2,3)	$\langle {\begin{pmatrix}1&1\\0&1\\\end{pmatrix}}\rangle$	cyclic group:Z3	3	8	1	4	4	--	--	3-Sylow
Z6 in SL(2,3)	$\langle {\begin{pmatrix}1&1\\0&1\\\end{pmatrix}},{\begin{pmatrix}-1&0\\0&-1\\\end{pmatrix}}\rangle$	cyclic group:Z6	6	4	1	4	4	--	--	3-Sylow normalizer
whole group	all elements	special linear group:SL(2,3)	24	1	1	1	1	trivial group	0
Total (7 rows)	--	--	--	--	7	--	15	--	--	--

1 What is the relationship between special linear group:SL(2,3) and alternating group:A4?

	The latter occurs both as a subgroup and as a quotient group of the former.
	The latter is isomorphic to a subgroup of index two in the former, but does not occur as a quotient of the former.
	The latter is isomorphic to a quotient of the former by a subgroup of order two, but does not occur as a subgroup.
	The latter occurs neither as a subgroup nor as a quotient group of the former.

2 For which of the following divisors of 24 does there not exist a subgroup of $SL(2,3)$ of that order?

	2
	3
	4
	6
	8
	12

3 Which of the following is correct about $SL(2,3)$?

	It is a direct product of its 2-Sylow subgroup and 3-Sylow subgroup
	It is a semidirect product of its Sylow subgroups, where the 2-Sylow subgroup is the normal piece and the 3-Sylow subgroup is the non-normal piece.
	It is a semidirect product of its Sylow subgroups, where the 3-Sylow subgroup is the normal piece and the 2-Sylow subgroup is the non-normal piece.
	Neither the 2-Sylow subgroups nor the 3-Sylow subgroups are normal.

4 Which of these groups of order 8 is the 2-Sylow subgroup of $SL(2,3)$ isomorphic to?

	cyclic group:Z8
	direct product of Z4 and Z2
	dihedral group:D8
	quaternion group
	elementary abelian group:E8

5 What is the order of the center of $SL(2,3)$?

	1
	2
	3
	4
	6