show that if H is a p sylow subgroup of a finite group G then for...

show that if H is a p sylow subgroup of a finite group G then for an arbitrary x in G x^-1 H x is also a p sylow subgroup of G

Expert Solution

Colby Messinger answered 2 years ago

Let G be a finite group and H a subgroup of G. Let a be an...

Let G be a finite group and H a subgroup of G. Let a be an element of G and aH = {ah : h is an element of H} be a left coset of H. If b is an element of G as well and the intersection of aH bH is non-empty then aH and bH contain the same number of elements in G. Thus conclude that the number of elements in H, o(H), divides the number of elements...

Let P be a finite p-group. Show that Φ(P) is the unique normal subgroup of P...

Let P be a finite p-group. Show that Φ(P) is the unique normal subgroup of P minimal such that the corresponding factor group is elementry abelian

4.- Show the solution: a.- Let G be a group, H a subgroup of G and...

4.- Show the solution: a.- Let G be a group, H a subgroup of G and a∈G. Prove that the coset aH has the same number of elements as H. b.- Prove that if G is a finite group and a∈G, then |a| divides |G|. Moreover, if |G| is prime then G is cyclic. c.- Prove that every group is isomorphic to a group of permutations. SUBJECT: Abstract Algebra (18,19,20)

Let G be a nonabelian group of order 253=23(11), let P<G be a Sylow 23-subgroup and...

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.

(Modern Algebra) Show that if G is a finite group that has at most one subgroup...

(Modern Algebra) Show that if G is a finite group that has at most one subgroup for each divisor of its order then G is cyclical.

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| = paq where p and q are primes and q < p. (c) |G| = 4p where p is a prime greater than four.

G is a group and H is a normal subgroup of G. List the elements of...

G is a group and H is a normal subgroup of G. List the elements of G/H and then write the table of G/H. 1. G=Z10, H= {0,5}. (Explain why G/H is congruent to Z5) 2. G=S4 and H= {e, (12)(34), (13)(24), (14)(23)

Let G be a group and K ⊂ G be a normal subgroup. Let H ⊂...

Let G be a group and K ⊂ G be a normal subgroup. Let H ⊂ G be a subgroup of G such that K ⊂ H Suppose that H is also a normal subgroup of G. (a) Show that H/K ⊂ G/K is a normal subgroup. (b) Show that G/H is isomorphic to (G/K)/(H/K).

Given a group G with a subgroup H, define a binary relation on G by a...

Given a group G with a subgroup H, define a binary relation on G by a ∼ b if and only if ba^(-1)∈ H. (a) (5 points) Prove that ∼ is an equivalence relation. (b) (5 points) For each a ∈ G denote by [a] the equivalence class of a and prove that [a] = Ha = {ha | h ∈ H}. A set of the form Ha, for some a ∈ G, is called a right coset of H...

(a) Suppose K is a subgroup of H, and H is a subgroup of G. If...

(a) Suppose K is a subgroup of H, and H is a subgroup of G. If |K|= 20 and |G| = 600, what are the possible values for |H|? (b) Determine the number of elements of order 15 in Z30 Z24.

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