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

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 sampling. 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... Please help! Thank you!!

Explanation / Answer

group is a p group iff it has prime order. sylow theorem :- if any other q | o(G) => by sylow, there exists an element with order q. So, 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

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