Use Exercises 3 and 7 to obtain a proof of Theorem 2 2 which is independent of Theorem 2 1 Data from Exercise 3 Suppose G is a finite abelian p group (Exercise 7) and x G has maximal order If Y G (x) has order p r , then there is a representative y G of the coset such that y p r Data from Exercise ...

Theorem 22 states that every finite abelian group G is a direct sum of cyclic groups We can prove th ...

Use Exercises 3 and 7 to obtain a proof of Theorem 2.2 which is independent of Theorem

Question:

Use Exercises 3 and 7 to obtain a proof of Theorem 2.2 which is independent of Theorem 2.1.

Data from Exercise 3

Suppose G is a finite abelian p-group (Exercise 7) and x ϵ G has maximal order. If Y ϵ G/(x) has order p^r, then there is a representative y ϵ G of the coset ȳ such that |y| = p^r.

Data from Exercise 7

A (sub)group in which every element has order a power of a fixed prime p is called a p-(sub)group.(a) G(p) is the unique maximum p-subgroup of G (that is, every p-subgroup of G is contained in G(p)).(b) where the sum is over all primes p such that G(p) ≠ 0.