There are some group properties which, if they are true in G / H and in H, must

Question

There are some group properties which, if they are true in G / H and in H, must be true in G. Here is a sample: Let G be a group, and H a normal subgroup of G. Prove the following... Let p be a prime number. If G / H and H are p groups, then G is a p group. A group G is called a p group if the order of every element x in G is a power of p. PROVE & SHOW ALL STEPS

Explanation / Answer

Actually a group is a p group iff it has prime order. The definition given is also equivalent by the sylow theorem - if any other q | o(G) => by sylow, there exists an element with order q. So, now we have the theorem : o(G) = o(G/H)o(H). So, now if both G/H and H are p groups, their order are p^a and p^b. So, o(G) = p^(a+b) : so, G is a p group Hence proved Message me if you have any doubts.

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