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

Cauchy's Theorem states that if prime p divides |G|, then G must contain at least one element whose order is exactly p.

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

An isomorphism is a bijective homomorphism φ: G → H; its existence guarantees that G and H are structurally identical as groups.

Question 3

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

Accepted Answer

2

Answer

A homomorphism maps the generator of Z_4 to an element whose order divides gcd(4, 6) = 2; only elements 0 and 3 in Z_6 have order dividing 2, giving exactly 2 homomorphisms.

Question 4

A group G is called solvable if:

Accepted Answer

It has a subnormal series with abelian factor groups

Answer

G is solvable if there exists a subnormal series 1 = G_0 ⊲ G_1 ⊲ ⋯ ⊲ G_n = G where each successive quotient G_{i+1}/G_i is abelian.

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

A Sylow p-subgroup has order p^k, where p^k is the exact highest power of the prime p dividing the group order |G|.

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Abstract Algebra and Group Theory 3