Center of general linear group:GL(2,3)
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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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.
|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 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|