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Abstract Algebra Question involving Quotient Groups?
Question:
Assuming that N ⊲ G, prove that if [G:N] is a prime, then G/N is cyclic. Is the converse true?
Thank you.
1 Answer
- DavidLv 71 decade agoFavorite Answer
any group of prime order is cyclic.
why? suppose |G| = p, prime. since |g| divides |G| for all g in G, |g| must either be 1 or p.
if |g| = 1, g = e, otherwise <g> has p elements, which must therefore be all of G.
the converse is NOT true. Z/nZ is cyclic for EVERY number n, including non-primes.
