Quiz:Alternating group:A4

This quiz is about alternating group:A4, which we take as the alternating group on the set $\! \{ 1,2,3,4 \}$ of size four.

Elements

See element structure of alternating group:A4 for background information.

Element order and conjugacy class structure

Review the conjugacy class structure: [SHOW MORE]

1 What are the possible orders of elements in the alternating group of degree four?

 1,2,3 only 1,3,4 only 1,2,3,4 only 1,2,3,6 only 1,2,3,4,6 only 1,2,3,4,6,12 only

2 Which of the following elements is in the alternating group on the set $\{ 1,2,3,4 \}$ and is not conjugate to its inverse in the alternating group? $()$ -- the identity element $(1,2)$ -- a 2-transposition $(1,2)(3,4)$ -- a double transposition $(1,2,3)$ -- a 3-cycle $(1,2,3,4)$ -- a 4-cycle

3 How many conjugacy classes of elements are there in the alternating group of degree four?

 1 2 3 4 5 6 7

4 How many orbits are there in the alternating group of degree four under the action of its automorphism group?

 1 2 3 4 5 6 7

Subgroups

See subgroup structure of alternating group:A4 for background information.

Basic stuff

Summary table on the structure of subgroups: [SHOW MORE]

View the lattice of subgroups as a picture: [SHOW MORE]

1 Which of the following factors of 12 does not occur as the order of a subgroup of the alternating group of degree four?

 1 2 3 4 6

2 Which of the following groups is not isomorphic to any subgroup of the alternating group of degree four?

 trivial group cyclic group:Z2, the cyclic group of order two cyclic group:Z3, the cyclic group of order three cyclic group:Z4, the cyclic group of order four Klein four-group, the non-cyclic group of order four