Center of general linear group:GL(2,3)
From Groupprops
This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z2 and the group is (up to isomorphism) general linear group:GL(2,3) (see subgroup structure of general linear group:GL(2,3)).
The subgroup is a normal subgroup and the quotient group is isomorphic to symmetric group:S4.
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Definition
is the general linear group of degree two over field:F3. In other words, it is the group of invertible matrices with entries over the field of three elements. The field has elements with .
is the subgroup:
is isomorphic to cyclic group:Z2. It is the center of (see center of general linear group is group of scalar matrices over center). The quotient group is (the projective general linear group of degree two over field:F3) which is isomorphic to symmetric group:S4.
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of the whole group | 48 | order of is . Here . |
order of the subgroup | 2 | As center: it has order . |
index of the subgroup | 24 | Follows from Lagrange's theorem. Also, index equals order of , which is . |
size of conjugacy class = index of normalizer | 1 | center is normal |
number of conjugacy classes in automorphism class | 1 | center is characteristic |
Subgroup-defining functions
Subgroup-defining function | What it means in general | Why it takes this value |
---|---|---|
center | set of elements that commute with every group element | center of general linear group is group of scalar matrices over center -- here the underlying ring is a field, which is commutative, so this just gives that the center is the group of scalar matrices over the multiplicative group of the field. |
third derived subgroup | derived subgroup of derived subgroup of derived subgroup | derived subgroup is SL(2,3) in GL(2,3), second derived subgroup is Q8 in GL(2,3) |
socle | join of all minimal normal subgroups | It is the unique minimal normal subgroup. The group is a monolithic group. |
Frattini subgroup | intersection of all maximal subgroups |