# Quiz:Subgroup structure of special linear group:SL(2,3)

### 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.

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

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

 1 2 3 4 6