Clarification of a statement of a theorem in group theory Let $n$ be a positive integer. A group $G$ is called $n$-abelian if $(xy)^n=x^ny^n$ for every $x,y\in G$.
The following corollary appears in the paper "A Classification Of $n$-Abelian Groups" by J.L.ALPERIN:
COROLLARY $1$ A finite group is $n$-abelian if, and only if, it is a homomorphic image of a subgroup of the direct product of a finite abelian group, a finite group of exponent dividing $n$ and a finite group of exponent dividing $n—1$.
I do not understand this statement. Does this corollary mean that $G$ is $n$-abelian iff $G\cong A\times A_n\times A_{n-1}$, where $A$ is a finite abelian group, $A_n$ a finite group of exponent dividing $n$ and $A_{n-1}$ a finite group of exponent dividing $n—1$? What does he mean by "homomorphic image"?
Remark: I have only a basic background in finite groups, so his proof is beyond my understanding.
 A: Alperin's corollary is somewhat opaque and not as strong as you might think. A group G is $n$-abelian if and only if the following holds: There exist an abelian group $A$, a group $P$ of exponent dividing $n$ and a group $Q$ of exponent dividing $n-1$. There exists a subgroup $H$ of the direct product $A\times P\times Q$ and a surjective homomorphism $H\to G$ (in other words, $G$ is (isomorphic to) a quotient of $H$).
Note that $H$ itself it not necessarily a direct product of subgroups of $A,P,Q$ respectively (look up Gorsat's theorem for a classification of subgroups of direct products). Also a quotient of $H$ is not necessarily isomorphic to a subgroup of $H$. In summary, it is quite difficult to pin down the structure of $G$ from such a description. Some authors would say that $G$ is a section of $A\times P\times Q$ or that $G$ is involved in such a group.
EDIT: More details: As every finite abelian group, $A$ is a direct product of cyclic groups of prime power order. We may assume that those prime powers do not divide $n(n-1)$, since otherwise we may add the corresponding factors to $P$ or $Q$ respectively. If $|A|$ is now coprime to $n(n-1)$, then $H$ is indeed of the form $A_1\times P_1\times Q_1$ with $A_1\le A$, $P_1\le P$ and $Q_1\le Q$ (this follows from Gorsat's theorem mentioned above). In general however, $A$ can "interfere" with $P$ or $Q$. As a concrete example, take $n=4$, $A=C_8$ (cyclic group of order $8$) and $P=D_8$ (dihedral group of order $8$). Then $A\times P$ has a non-abelian subgroup $H$ of order $32$ and exponent $8$ which cannot be written as $A_1\times P_1$. Moreover, $H$ has a quotient isomorphic to the quaternion group $Q_8$. This can be checked most conveniently by computer, for instance, with GAP.
