The discussion revolves around proving that the set H of all solutions satisfying the equation x^n = e forms a subgroup of an abelian group G with identity e. Participants are exploring the implications of the group being abelian and questioning the necessity of this condition in the proof.
There is an ongoing exploration of the relationship between the abelian property of group G and the subgroup H. Some participants express confusion about why G must be abelian, while others reference previous responses that address this concern. The discussion is productive, with participants engaging in practice problems to reinforce their understanding.
Participants note that if G is not abelian, the equality (ab)^n = a^n b^n may not hold, which is a critical point in the discussion of the subgroup properties.
if a^n=e and b^n= e => (ab)^n=e so ab belongs to H.
where does the abelian part come in?
That was answered in ircdan's first response. If G is not abelian, then (ab)n is NOT, in general, anbn so you cannot use the fact that an= bn= e.rsa58 said:by the way i still don't understand why G must be abelian in the first problem.