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

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

Elements

Quiz:Element structure of 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]

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.

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

1
2
3
4
6