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Math Help - Endomorphisms

  1. #1
    Super Member Aryth's Avatar
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    Endomorphisms

    I have been given a problem where I have to find the group of all endomorphisms on \mathbb{Z}_4, Hom(\mathbb{Z}_4). I have only been able to come up with f(a) = a^n where a\in \mathbb{Z}_4 and n is an integer greater than or equal to 0.

    I guess the questions I'm asking are:

    - Are there any more?
    - How do I know if I have found them all?
    - Is there a way to go about finding all the endomorphisms?
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  2. #2
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    Re: Endomorphisms

    I will assume you mean abelian group endomorphisms, and not ring endomorphisms. I will write Z4 additively.

    As a group, Z4 is generated by 1, so any endomorphism is completely determined by the image of 1. We have 4 choices for such an endomorphism f:

    f(1) = 0. This is the trivial map, which sends everything to 0.

    f(1) = 1. This is the identity map, which is an automorphism.

    f(1) = 2. This is a quotient map onto the subgroup {0,2} (which sends a to 2a (mod 4)).

    f(1) = 3. This is another automorphism, which sends a to -a.

    So Hom(Z4,Z4) has 4 elements, and in fact is isomorphic to the multiplicative monoid of Z4.

    By the way, this is NOT a group, only the automorphisms Aut(Z4) form a group, of order 2. The endomorphisms: a-->0, a --> 2a, are not invertible.
    Last edited by Deveno; January 20th 2014 at 03:44 PM.
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