Order has only two prime factors implies prime divisor with larger class two subgroups is core-nontrivial

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Statement

Suppose p and q are two distinct primes, and G is a group of order p^aq^b for nonnegative integers a,b. Suppose e_p(G) denotes the maximum of the orders of p-subgroups of G of nilpotence class two. Then, if e_p(G) > e_q(G), p is a core-nontrivial prime divisor of G: G has a normal p-subgroup.

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