# Quiz:Subgroup structure of alternating group:A5

See subgroup structure of alternating group:A5 for background information.

### Basic stuff

1 For which of the following divisors of 60 does there exist a subgroup of the alternating group of degree five with that as index?

 2 3 4 5

2 What is the smallest possible order of a group that is not isomorphic to any subgroup of the alternating group of degree five?

 2 3 4 5 6

3 For which of the following pairs of primes does the alternating group of degree five have a Hall subgroup corresponding to that pair of primes?

 All pairs of primes, i.e., (2,3), (2,5), and (3,5) (2,3) and (2,5) only (2,3) only (2,3) and (3,5) only (3,5) only (2,5) only

4 The alternating group of degree five has a maximal subgroup whose index is not a prime number. What is the best description of this subgroup?

 Cyclic of order five Dihedral of order ten, degree five Cyclic of order ten Alternating of degree four Cyclic of order fifteen

5 The alternating group of degree five can be expressed as a product of two subgroups $HK$ where $H,K$ are permutable complements. What are the orders of these subgroups?

 2 and 30 3 and 10 4 and 15 5 and 12 6 and 10