Suppose p is a prime and C is a cyclic group of order p3 Sho

Suppose p is a prime and C is a cyclic group of order p^3. Show that C cannot be written as a non-trivial direct product C = H times K, with H and K subgroups of (7 and \\H\\, \\K\\ > 1.

Solution

Since p is a prime and G is a cyclic group of order p³ 0_G is the only element of G with finite order

If p0 and pg=0G, then p(g,0H)=0G×H contradicting the fact that QQ has no non-zero elements of finite order

Hence G cannot be writtwn as a nontrivial direct product G=H x K.

If hH and nN^*, there exist (h₀,k₀)H×K such that hn=h₀+k₀; so nk₀=hnh₀HK={0}, hence k₀=0 and h/nH