Let ϕ∈End(G) s.t. ∃n≥0,ker(ϕn)=G If kerϕ or [G:Inϕ] is finite, then G is finite.

Carla Murphy

Carla Murphy

Answered question


ϕEnd(G) s.t.
If kerϕ or [G:Inϕ] is finite, then G is finite.

Answer & Explanation



Beginner2022-01-13Added 33 answers

Step 1
Assume that ϕ has finite kernel. Consider the map
It has finite kernel. So if it has finite range, it has finite domain. By induction, if ker(ϕ) is finite, so is every ker(ϕn) and in particular G.
If the image of ϕ has finite index, then ϕk(ϕ(G)) has finite index in ϕk(G) for all k. It follows that 1=ϕn(G) has finite index in G so that G is finite.


Beginner2022-01-14Added 40 answers

Step 1
For the case where kerϕ is finite:
Note that if ker(ϕ)KH, then
by the isomorphism theorems.
Now, ϕ(kerϕi+1)kerϕi
Thus, in particular,
So if kerϕ is finite, then so is ker(ϕ2). Now proceed by induction to show that kerϕr is finite.
If Im(ϕ) has finite index, we can proceed similarly. Note that for arbitrary H and K, we know that
So the I,(ϕ2) also has finite index. Proceed by induction to conclude that G is finite.

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