The discussion revolves around a problem in group theory concerning a subgroup H of prime index in a finite group G. Participants explore the implications of this condition on the normalizer N(H) of the subgroup, specifically whether N(H) equals G or H.
Participants express various viewpoints and uncertainties regarding the implications of the subgroup's prime index and the properties of the normalizer. No consensus is reached on the next steps or conclusions.
Participants acknowledge the need to clarify the definitions and relationships between the orders of the groups involved, as well as the implications of normality in the context of the problem.
Well as H has prime index it means it is cyclic. Also we have that p=ord(H) | ord(N) | ord(G). So we should have ord(N)=##k_1##p and ord(G)=##k_2##ord(N)=##k_1## ##k_2## p. So we have to show that either ##k_1##=1 or ##k_2##=1. This is what I got by now. But I don't know what to do nextfresh_42 said:What do you know about the index of a subgroup? And where is ##N_G(H)## settled with respect to ##H## and ##G##?
So p=ord(H) | ord(N) | ord(G) means ##H \leq N_G(H) \leq G##. The index of ##H## is also equal to ##|G/H|##. Do you know what happens, if you factorize ##H## across these inclusions?Silviu said:Well as H has prime index it means it is cyclic. Also we have that p=ord(H) | ord(N) | ord(G). So we should have ord(N)=##k_1##p and ord(G)=##k_2##ord(N)=##k_1## ##k_2## p. So we have to show that either ##k_1##=1 or ##k_2##=1. This is what I got by now. But I don't know what to do next
Sorry I am a little confused. You mean something like ##|H/H| \le |N(H)/H| \le |G/H| ##? This means ##1 \le |N(H)/H| \le |G/H|##. So what next?fresh_42 said:So p=ord(H) | ord(N) | ord(G) means ##H \leq N_G(H) \leq G##. The index of ##H## is also equal to ##|G/H|##. Do you know what happens, if you factorize ##H## across these inclusions?
Thank you so much! I kept reading a subgroup of prime order. It is index not order. Now it is easy. Sorry for this and thank you again!fresh_42 said:If ##|G/H|=p## prime, how many orders fit in between ##1## and ##p##?
The essential part to be confused of, is the fact, that these cosets are possible disregarding whether ##H## is normal or not. Do you know why?