Question

In: Advanced Math

Prove that an abelian group G of order 2000 is the direct product PxQ where P...

Prove that an abelian group G of order 2000 is the direct product PxQ where P is the Sylow-2 subgroup of G, and Q the Sylow-5 subgroup of G. (So order of P=16 and order or Q=125).

Expert Solution

Note that . Thus by Sylow theorem there exists a Sylow-2 subgroup of order and Sylow-5 subgroup Q of order . Since G is commutative to show G is a direct product of P and Q it is enough to show and , where e is the identity of G.

Let , then note that since each P and Q are finite group we have and , this gives us . Note note that , hence .

Hence .

Note that , since G is a finite group to show , it is enough to show .

Note that if for pair in , if we can show , then in this way we can definie a bijection from to , by sending each pair . Thus we will have . Now note that . Hence we will get . So to show it is enough to show => .

Proof by contradiction. Suppose we have . Now note that thus we have the element . Since we have proved hence we get , which contradicts the fact . Hence done.

Colby Messinger answered 2 years ago

prove that a group of order 9 is abelian

abstract algebra Let G be a finite abelian group of order n Prove that if d...

abstract algebra Let G be a finite abelian group of order n Prove that if d is a positive divisor of n, then G has a subgroup of order d.

Let (G,+) be an abelian group and U a subgroup of G. Prove that G is...

Let (G,+) be an abelian group and U a subgroup of G. Prove that G is the direct product of U and V (where V a subgroup of G) if only if there is a homomorphism f : G → U with f|U = IdU

(a) Let G be a finite abelian group and p prime with p | | G...

(a) Let G be a finite abelian group and p prime with p | | G |. Show that there is only one p - Sylow subgroup of G. b) Find all p - Sylow subgroups of (Z2500, +)

Theorem 2.1. Cauchy’s Theorem: Abelian Case: Let G be a finite abelian group and p be...

Theorem 2.1. Cauchy’s Theorem: Abelian Case: Let G be a finite abelian group and p be a prime such that p divides the order of G then G has an element of order p. Problem 2.1. Prove this theorem.

Let G be an abelian group and n a fixed positive integer. Prove that the following...

Let G be an abelian group and n a fixed positive integer. Prove that the following sets are subgroups of G. (a) P(G, n) = {gn | g ∈ G}. (b) T(G, n) = {g ∈ G | gn = 1}. (c) Compute P(G, 2) and T(G, 2) if G = C8 × C2. (d) Prove that T(G, 2) is not a subgroup of G = Dn for n ≥ 3 (i.e the statement above is false when G is...

Let G be a group of order p am where p is a prime not dividing...

Let G be a group of order p am where p is a prime not dividing m. Show the following 1. Sylow p-subgroups of G exist; i.e. Sylp(G) 6= ∅. 2. If P ∈ Sylp(G) and Q is any p-subgroup of G, then there exists g ∈ G such that Q 6 gP g−1 ; i.e. Q is contained in some conjugate of P. In particular, any two Sylow p- subgroups of G are conjugate in G. 3. np ≡...

Assume G is an Abelian group of order 144 and G contains at least 5 elements...

Assume G is an Abelian group of order 144 and G contains at least 5 elements of order 72. (a) Determine the possible structures of G (i.e., write it as a direct product of cyclic groups). If there are more than one structures, list them all. (b) For each isomorphic class of G, how many elements of order 12 does it have?

if order of a= order of b in a finite Abelian group , is order of...

if order of a= order of b in a finite Abelian group , is order of ab= order of a?

Show that if G is a group of order np where p is prime and 1...

Show that if G is a group of order np where p is prime and 1 < n < p, then G is not simple. (Please do not use Sylow theorem)