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Abstract Algebra

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Group of Given Order: Properties of a Group of Order $15$

ABSALG-PHBHKA

Which of the following are NOT properties of every group $G$ of order $15$?

Notation: We let $e$ be the neutral (identity) element of $G$ and $\circ$ the composition law for $G$.

Select ALL that apply.

A

$G$ is generated by any $g\in G$ with $g\not=e$.

B

$G$ is cyclic.

C

$G$ is abelian.

D

$G$ is simple.

E

$G$ is unique up to group isomorphism.