Groups Quiz

G is a finite group. The subgroups of G are {a,c}, {b,c,e}, {c,d,e,f,g}. The identity element of G is:

• It is not possible to say.

• a

• c

• e

G is a finite group. The subgroups of G are {a,c}, {b,c,e}, {c,d,e,f,g}. The smallest possible value of the order of group G is:

• 5

• 8

• 15

• 30

Which of the following are subgroups of the group of integers under addition?

A: the set of odd integers under addition

B: the set of even integers under addition

• A only

• B only

• A and B

• neither

 $S_4$  is called the permutation group. It is the group of all possible permutation of the numbers [1234]. Permutations are combined in the same way as functions: you carry out one permutation after another.

The number of subgroups of  $S_4$  which have order 2 is

• 3

• 6

• 9

• 12

 $S_4$  is called the permutation group. It is the group of all possible permutation of the numbers [1234]. Permutations are combined in the same way as functions: you carry out one permutation after another.

The possible orders of proper subgroups of  $S_4$  are

• 2,3,4,6

• 1,2,3,4,6,8,12,24

• 2,3,4,6,8,12

• 1,2,3,4,6,8,12

 $S_4$  is called the permutation group. It is the group of all possible permutation of the numbers [1234]. Permutations are combined in the same way as functions: you carry out one permutation after another.

Which of the following are subgroups of  $S_4$  ?
A: {[1234],[2143],[3412],[4321]}
B: {[1234],[1243],[2134],[2143]}

• A only

• B only

• A and B

• neither

Which of these groups is cyclic?

• A and B

• A only

• B only

• neither

In the cyclic group, how many different generators of the group are there?

• 1

• 2

• 3

• 5

In group A, what is the order of element s?

• 2

• 3

• 4

• 6

In group B, what is the order of element m?

• 1

• 2

• 3

• 6