Groups

Some cool results

• Let \(G\) be a group such that \(a^2 = 1\) for every \(a \in G,\) then \(G\) is abelian
• Let \(G\) be a group of order \(p^n,\) where \(p\) is a prime and \(n \in \mathbb{Z}^+.\) Then, \(Z(G)\) is nontrivial.
• Any group of order \(p^2\) for prime \(p\) is abelian.
• Any group of order \(pq\) where \(p \nmid q - 1\) is cyclic. (Of course, \(p\) and \(q\) are primes as usual.)
• If \(G/Z(G)\) is cyclic, then \(G\) is abelian.

Sylow

• This is where I’ve posted most of content related to groups.
  I have classified groups of order \(n\) on the basis of their simplicity for \(n \le 200.\)
• Exercise 4.5 of Dummit Foote.