I'm translating a paper from the 70's where the author cites László Fuchs' Abelian groups (the entire book, no page number). As you can imagine, given the size and differences in editions of Fuchs' book, I am having trouble connecting the dots.
Here is the situation: $G$ is an infinite, abelian, and locally finite group. $H$ is an infinite subgroup of $G$ which is of finite index in $G$. From this, the step in a proof I am reading continues this way
... we deduce the socle of $G$ is of finite length and that $G$ is an Artinian group.
The justification of the block above is what I am asking about.
In the newest edition (2015) of Fuchs' Abelian groups, I feel think the relevant theorem is this:
Theorem 5.3 (Prufer, Kurosh, Yahya)
(i) $A$ is finitely cogenerated
(ii) $A$ is an essential extension of a finite group
(iii) $A$ is torsion of finite rank
(iv) $A$ is a direct sum of a finite number of cocyclic groups
(v) the subgroups of $A$ satisfy the minimum condition
[later] Obeserve that (ii) is equivalent to the finiteness of the socle in a torsion group.
This is the very first theorem in the section, preceded only by the definition of finite cogeneration.
The local finiteness obviously makes $G$ torsion, so I can see why both conclusions are linked... but how do you use the fact that $|G:H|$ is finite to prove one of these conditions?
I feel like I'm overlooking some connection between $G/H$ and $G$, possibly about the socles. In general module theory, there usually isn't a connection between the two, but perhaps since $G$ is locally finite there is a connection in my blind spot.
I've decided the original source is in order:
$A$ is a right self-injective ring, and $G$ is, as proven in an earlier step, at least a locally finite group. The $G_1$ and $H_1$ in this snippet are the $G$ and $H$ I was referring to in my original description. The citation (7) refers to Fuchs' Abelian groups. The theorem of Faith has to do with the injectivity of a free $A[H_1]$ module. Basically it allows you to conclude that $A[G_1]$ is a direct sum of finitely many copies of $A[H_1]$, and this means $|G_1:H_1|$ is finite.
Perhaps the theorem that's needed (which did not come out in my description above) is that if $G$ is a locally finite, infinite abelian group whose infinite subgroups are all of finite index, $G$ is Artinian? If this is the case, then a reference to that result would be an acceptable solution to this problem. (Hopefully in Fuchs, but elsewhere would be fine.)