Subgroup structure of special linear group:SL(2,5)

This article gives specific information, namely, subgroup structure, about a particular group, namely: special linear group:SL(2,5).
View subgroup structure of particular groups | View other specific information about special linear group:SL(2,5)

This article describes the subgroup structure of special linear group:SL(2,5), which is the special linear group of degree two over field:F5. The group has order 120.

Family contexts

Family name	Parameter values	General discussion of subgroup structure of family
special linear group of degree two	field:F5, i.e., the group $SL(2,5)$	subgroup structure of special linear group of degree two over a finite field
double cover of alternating group $2\cdot A_{n}$	degree $n=5$, i.e., the group $2\cdot A_{5}$	subgroup structure of double cover of alternating group

Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
normal Hall implies permutably complemented, Hall retract implies order-conjugate

Quick summary

Item	Value
number of subgroups	76
number of conjugacy classes of subgroups	12
number of automorphism classes of subgroups	12
isomorphism classes of Sylow subgroups, corresponding fusion systems, and Sylow numbers	2-Sylow: quaternion group (order 8) with its non-inner fusion system (see non-inner fusion system for quaternion group), Sylow number 5 3-Sylow: cyclic group:Z3 with its non-inner fusion system, Sylow number 10 5-Sylow: cyclic group:Z5, Sylow number 6
Hall subgroups	Other than the whole group, trivial subgroup, and Sylow subgroups, there is a $\{2,3\}$-Hall subgroup of order 24 (SL(2,3) in SL(2,5)). There is no $\{2,5\}$-Hall subgroup or $\{3,5\}$-Hall subgroup.
maximal subgroups	There are maximal subgroups of order 12 (index 10), order 20 (index 6) and order 24 (index 5).
normal subgroups	The only proper nontrivial normal subgroup is center of special linear group:SL(2,5), which is isomorphic to cyclic group:Z2 and the quotient group is isomorphic to alternating group:A5.

Defining functions

Subgroup-defining functions and associated quotient-defining functions

Subgroup-defining function	What it means	Value as subgroup	Value as group	Order	Associated quotient-defining function	Value as group	Order (= index of subgroup)
center	elements that commute with every group element	center of special linear group:SL(2,5)	cyclic group:Z2	2	inner automorphism group	alternating group:A5	60
derived subgroup	subgroup generated by all commutators	whole group	special linear group:SL(2,5)	120	abelianization	trivial group	1
perfect core	the subgroup at which the derived series stabilizes	whole group	special linear group:SL(2,5)	120	?	trivial group	1
hypocenter	the subgroup at which the lower central series stabilizes	whole group	special linear group:SL(2,5)	120	?	trivial group	1
hypercenter	the subgroup at which the upper central series stabilizes	center of special linear group:SL(2,5)	cyclic group:Z2	2	?	alternating group:A5	60
Frattini subgroup	intersection of all maximal subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	Frattini quotient	alternating group:A5	60
Jacobson radical	intersection of all maximal normal subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	?	alternating group:A5	60
socle	join of all minimal normal subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	socle quotient	alternating group:A5	60
Baer norm	intersection of normalizers of all subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	?	alternating group:A5	60
join of all abelian normal subgroups	subgroup generated by all the abelian normal subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	?	alternating group:A5	60
Fitting subgroup	subgroup generated by all the nilpotent normal subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	?	alternating group:A5	60
solvable radical	subgroup generated by all the solvable normal subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	?	alternating group:A5	60
socle over solvable radical	its quotient by the solvable radical of the group is the socle of the quotient of the group by the solvable radical	whole group	special linear group:SL(2,5)	120	?	trivial group	1
permutation kernel	too complicated to describe here	whole group	special linear group:SL(2,5)	120	?	trivial group	1

Sylow subgroups

Compare and contrast with subgroup structure of special linear group of degree two over a finite field#Sylow subgroups

We are considering the group $SL(2,q)$ with $q=p^{r}$ a prime power, $q=5,p=5,r=1$. The prime $p=5$ is the characteristic prime.

Sylow subgroups for the prime 5

The prime 5 is the characteristic prime $p$, so we compare with the general information on $p$-Sylow subgroups of $SL(2,q)$.

Item	Value for $SL(2,q)$, generic $q$	Value for $SL(2,5)$ (so $q=p=5,r=1$)
order of $p$-Sylow subgroup	$q$	5
index of $p$-Sylow subgroup	$q^{2}-1$	24
explicit description of one of the $p$-Sylow subgroups	unitriangular matrix group of degree two: $\{{\begin{pmatrix}1&b\\0&1\\\end{pmatrix}}\mid b\in \mathbb {F} _{q}\}$	See 5-Sylow subgroup of special linear group:SL(2,5)
isomorphism class of $p$-Sylow subgroup	additive group of the field $\mathbb {F} _{q}$, which is an elementary abelian group of order $p^{r}$	cyclic group:Z5
explicit description of $p$-Sylow normalizer	Borel subgroup of degree two: $\{{\begin{pmatrix}a&b\\0&a^{-1}\\\end{pmatrix}}\mid a\in \mathbb {F} _{q}^{\ast },b\in \mathbb {F} _{q}\}$	See 5-Sylow normalizer of special linear group:SL(2,5)
isomorphism class of $p$-Sylow normalizer	It is the external semidirect product of $\mathbb {F} _{q}$ by the multiplicative group of $\mathbb {F} _{q}^{\ast }$ where the latter acts on the former via the multiplication action of the square of the acting element. For $p=2$ (so $q=2,4,8,\dots$), it is isomorphic to the general affine group of degree one $GA(1,q)$. For $q=3$, it is cyclic group:Z6 and for $q=5$, it is dicyclic group:Dic20.	dicyclic group:Dic20
order of $p$-Sylow normalizer	$q(q-1)=p^{2r}-p^{r}$	20
$p$-Sylow number (i.e., number of $p$-Sylow subgroups) = index of $p$-Sylow normalizer	$q+1$ (congruent to 1 mod p, as expected from the congruence condition on Sylow numbers)	6

Sylow subgroups for the prime 2

We are in the subcase where $\ell =2$ ($\ell$ being the prime for which we are taking Sylow subgroups) and $q\equiv 5{\pmod {8}}$.

Item	Value for $\ell =2,q\equiv 5{\pmod {8}}$	Value for $\ell =2,q=p=5,r=1$ (our case)
order of 2-Sylow subgroup	8	8
index of 2-Sylow subgroup	$(q^{3}-q)/8$	15
explicit description of one of the 2-Sylow subgroups	Since multiplicative group of a finite field is cyclic, $\mathbb {F} _{q}^{\ast }$ is cyclic of order $q-1$. Let $H$ be its unique subgroup of order 4. Then, the 2-Sylow subgroup is $\{{\begin{pmatrix}a&0\\0&a^{-1}\\\end{pmatrix}}\mid a\in H\}\cup \{{\begin{pmatrix}0&a\\-a^{-1}&0\\\end{pmatrix}}\mid a\in H\}$	2-Sylow subgroup of special linear group:SL(2,5)
isomorphism class of 2-Sylow subgroup	quaternion group	quaternion group
explicit description of 2-Sylow normalizer	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.	SL(2,3) in SL(2,5)
isomorphism class of 2-Sylow normalizer	special linear group:SL(2,3)	special linear group:SL(2,3)
order of 2-Sylow normalizer	24	24
2-Sylow number (i.e., number of 2-Sylow subgroups) = index of 2-Sylow normalizer	$(q^{3}-q)/24$	5

Sylow subgroups for the prime 3

Here, $\ell =3$ and we are interested in the $\ell$-Sylow subgroups.

We are in the subcase $\ell$ is an odd prime dividing $q+1$. Suppose $\ell ^{t}$ is the largest power of $\ell$ dividing $q+1$. In our case, $t=1$.

Item	Value for generic $\ell ,t,p,q,r$	Value for $\ell =3,t=1,p=q=5,r=1$
order of $\ell$-Sylow subgroup	$\ell ^{t}$	3
index of $\ell$-Sylow subgroup	$(q^{3}-q)/\ell ^{t}$	40
explicit description of one of the $\ell$-Sylow subgroups	Since multiplicative group of a finite field is cyclic, $\mathbb {F} _{q^{2}}^{\ast }$ is cyclic of order $q^{2}-1$. Further, via the action on a two-dimensional vector space over $\mathbb {F} _{q}$, we can embed $\mathbb {F} _{q^{2}}^{\ast }$ inside $GL(2,q)$. The image of the $\ell$-Sylow subgroup of $\mathbb {F} _{q^{2}}^{\ast }$ in $GL(2,q)$ actually lands inside $SL(2,q)$, and this image is a $\ell$-Sylow subgroup of $SL(2,q)$	3-Sylow subgroup of special linear group:SL(2,5)
isomorphism class of $\ell$-Sylow subgroup	cyclic group of order $\ell ^{t}$	cyclic group:Z3
explicit description of $\ell$-Sylow normalizer	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.
isomorphism class of $\ell$-Sylow normalizer	Case $p=2$: dihedral group of order $2(q+1)$ Case $p\neq 2$: dicyclic group of order $2(q+1)$	dicyclic group:Dic12
order of $\ell$-Sylow normalizer	$2(q+1)$	12
$\ell$-Sylow number (i.e., number of $\ell$-Sylow subgroups) = index of $\ell$-Sylow normalizer	$q(q-1)/2$ (congruent to 1 mod $\ell$, as expected from the congruence condition on Sylow numbers)	10