Let ϕ∈End(G) s.t. ∃n≥0,ker(ϕn)=G If kerϕ or [G:Inϕ] is finite, then G is finite.
If or is finite, then G is finite.
Answer & Explanation
Beginner2022-01-13Added 33 answers
Assume that has finite kernel. Consider the map
It has finite kernel. So if it has finite range, it has finite domain. By induction, if is finite, so is every and in particular G.
If the image of has finite index, then has finite index in for all k. It follows that has finite index in G so that G is finite.
Beginner2022-01-14Added 40 answers
For the case where is finite:
Note that if then
by the isomorphism theorems.
Thus, in particular,
So if is finite, then so is . Now proceed by induction to show that is finite.
If has finite index, we can proceed similarly. Note that for arbitrary H and K, we know that
So the also has finite index. Proceed by induction to conclude that G is finite.
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