your question doesn't make sense, as written. if H < K, then what does H/K mean?
Let be a finite group. Let and be normal -subgroups of for some prime with being a proper subgroup of and is elementary abelian p-group. Let be a normal subgroup of with . If contain every element of order of and , then contain every element of order of .
I need to prove the above statement. Thanks in advance.
I fix the mistake in the question with some additions.
Let be a finite solvable group. Let be a chief factor of that is not of prime order, where is a -subgroup of for some prime divides the order of . Let be a proper normal subgroup of with and . If contain every element of order of and , then contains every element of order of .
I need to prove the above statement.
Here is what I know.
Since is solvable then is abelian -group of exponent . is a normal subgroup of that contains . So H= .
Thanks in advance.