Show that the groups of the following orders have a normal Sylow subgroup. (a) |G| =...

Show that the groups of the following orders have a normal Sylow subgroup.

(a) |G| = pq where p and q are primes.

(b) |G| = p^aq where p and q are primes and q < p.

(c) |G| = 4p where p is a prime greater than four.

Expert Solution

Colby Messinger answered 2 years ago

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(a) Let G and G′ be finite groups whose orders have no common factors. Show that the only homomorphism φ:G→G′ is the trivial one. (b) Give an example of a nontrivial homomorphism φ for the given groups, if an example exists. If no such homomorphism exists, explain why. i.φ: Z16→Z7 ii.φ: S4→S5

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Let G be a nonabelian group of order 253=23(11), let P<G be a Sylow 23-subgroup and Q<G a Sylow 11-subgroup. a. What are the orders of P and Q. (Explain and include any theorems used). b. How many distinct conjugates of P and Q are there in G? n23? n11? (Explain, include any theorems used). c. Prove that G is isomorphic to the semidirect product of P and Q.

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Show that the groups of the following orders have a normal Sylow subgroup. (a) |G| =...

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