(a) Let G be a finite abelian p-group (Exercise 7). Show that for each n ≥ 0, pⁿ⁺¹G ∩ G[p] is a subgroup of pⁿG ∩ G[p].(b) Show that (pⁿG ∩ G[p])/(pⁿ⁺¹G ∩ G[p]) is a direct sum of copies of Z_p; let k be the number of copies.(c) Write G as a direct sum of cyclics; show that the number k of part (b) is the number of summands of order pⁿ⁺¹.

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.

Transcribed Image Text:

G = G(p),