Groups TT23, Classifying groups
Flashcards
What can you deduce about finite groups of size $p^2$ where $p$ prime?
$G$ is Abelian and isomorphic to $C _ p \times C _ p$ or $C _ {p^2}$.
What can you deduce about the centre of a group of size $p^n$ where $p$ is some prime?
They have a non-trivial centre, e.g.
\[ \vert Z(G) \vert > 1\]What are (up to isomorphism) the two groups of size $4$?
- $C _ 4$
- $V _ 4 \cong C _ 2 \times C _ 2$ (Klein-4 group)
What are (up to isomorphism) the five groups of size 8?
- $C _ 2 \times C _ 2 \times C _ 2$
- $C _ 2 \times C _ 4$
- $C _ 8$
- $D _ 8$
- $Q _ 8$
Proofs
@Prove that if $G$ is a finite group of order $p^2$, then $G$ is Abelian and isomorphic to $C _ p \times C _ p$ or $C _ {p^2}$ by appealing to the fact that any group of order $p^n$ has a non-trivial centre.
Todo.
@Prove that if the order of a finite group is $p^n$ then it has a non-trivial centre.
Todo.