Notes - Groups TT23, Classifying groups

Created: May 12, 2023

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.

\[|Z(G)| > 1\]

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 two groups of size $6$?
  • $C _ 6$
  • $S _ 3$

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 $p$, then $G$ is Abelian and isomorphic to $C _ p \times C _ p$ or $C _ {p^2}$ by appealing to the fact that the centre of any group 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.



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