BMath Bachelor of Mathematics Abstract Algebra and Group Theory 3

Question 1

Cauchy's Theorem guarantees that if a prime p divides |G| for a finite group G, then G contains:

Accepted Answer

An element of order p

Answer

A normal subgroup of order p

Answer

A unique subgroup of order p

Answer

An element of order p²

Question 2

Two groups G and H are isomorphic if and only if:

Accepted Answer

There exists a bijective homomorphism from G to H

Answer

They have the same number of elements

Answer

They have the same generators

Answer

They are both abelian

Question 3

How many distinct group homomorphisms are there from Z_4 to Z_6?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 4

A group G is called solvable if:

Accepted Answer

It has a subnormal series with abelian factor groups

Answer

It is abelian

Answer

Every subgroup of G is normal

Answer

It has prime order

Question 5

If p^k divides |G| but p^(k+1) does not, what is the order of a Sylow p-subgroup of G?

Accepted Answer

p^k

Answer

p

Answer

|G|/p

Answer

p^(k−1)

Question 6

Which of the following groups is an example of a simple group?

Accepted Answer

Z_5

Answer

Z_4

Answer

Z_6

Answer

S_4

Question 7

The commutator subgroup [G, G] (also called the derived subgroup) of G is the smallest normal subgroup N such that:

Accepted Answer

G/N is abelian

Answer

G/N is cyclic

Answer

N equals G

Answer

N is trivial