Advanced Group Theory MCQs

Let G be a finite group and p be a prime dividing |G|. Then which of the following statements is always true?

• G has a unique subgroup of order p

• G has a subgroup of order p

• G is cyclic

• Every subgroup of order p is normal

If G is a finite group and H is a subgroup of G with index 2, then:

• H is cyclic

• H is trivial

• H is normal in G

• H is the center of G

Let G be a finite group such that G/Z(G) is cyclic. Then G is:

• Simple

• Cyclic

• Abelian

• Nilpotent

In a finite group, the number of elements conjugate to an element g is equal to:

• Order of g

• Index of the centralizer C_G(g)

• Order of C_G(g)

• Order of Z(G)

Let G be a group of order 231. The number of Sylow 11-subgroups in G is:

• 1

• 3

• 7

• 21

The number of elements of order 5 in S5 is:

• 12

• 24

• 0

• 20

Let G be a group of order 56. The number of Sylow 7-subgroups in G can be:

• 1 or 8

• 2 or 4

• 1, 2, 4 or 8

• Only 1

Let G be a finite group of order p^n, where p is a prime. Then:

• G has a unique Sylow p-subgroup

• Every non-identity element has order p

• Z(G) ≠ {e}

• G is cyclic

No of generators in the group Z6 of addition modulo 6 are:

• 2

• 3

• 5

• 6

The number of elements of order 2 in Dihedral group of order 8 is:

• 2

• 3

• 5

• 4