I am quite new to group theory, so I am trying to get my head around Sylow's Theorems and other stuff....I got an exercice here and I am not sure how to go on with the proofs.

We have a group G of order pqr, where p, q and r are distinct primes such that q

a) Deduce that the commutator group C of G is part of P.

Now assume that p =/= 1 (mod r) and p=/= 1 (mod q).

b) Prove that the number of subgroups of r elements of G is 1 o pq, and prove that the number of subgroups of q elements of G is 1 o pr.

c) Prove that G doesn't contain simultaneously pq subgroups of r and pr subgroups of q. Deduce that G has a unique subgroup of order r or an unique subgroup of order q, and that in any case, that subgroup is normal. Give it the name K.

d) Prove that G/K is abelian. Deduce that C = (id) and that G is abelian.

My thoughts to the questions so far: a) Use Sylow's second theorem, because we can show that the commuter group is a conjugate to the Sylow subgroups? b) Just applying Sylow's third theorem or is there more to proof? c) For me, it's visually obvious, but I don't know how to proof it.

Any help is much appreciated!

  • $\begingroup$ If it's visually obvious to you then you are almost there, and you should try to find a rigorous proof yourself (this will help you much more than just reading the answer of someone else). $\endgroup$ – Dietrich Burde Apr 10 '15 at 12:17
  • $\begingroup$ @ramb Read your third line's last words: "...such that $\;q\;$ ..." ?? $\endgroup$ – Timbuc Apr 10 '15 at 13:07
  • $\begingroup$ And $P$ is undefined in the fourth line. Also, while it may not be an exact duplicate, this looks to be the same problem as math.stackexchange.com/questions/1215064 $\endgroup$ – Derek Holt Apr 10 '15 at 13:39

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