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Math Help - Abstract Alg.-Abelian groups presentation

  1. #1
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    Abstract Alg.-Abelian groups presentation

    Let Cn be a cyclic group of order n.
    A. How many sub-groups of order 4 there are in C2xC4... explain.
    B. How many sub-groups of order p there are in CpxCpxC(p^2) when p is a prime? explain.
    C. Prove that if H is cyclic of order 8 then Aut(H) is a non-cyclic group. WHAT is its order?
    D. What is the Automorphism group of an infinite cyclic group?

    About A-> it's obvious (because it has index 2) that a subgroup of order 4 is normal...But I can't figure out how many subgroups of this form there are...
    About the other parts-I've no idea...

    I'll be delighted to get guidance about all the parts in this question.

    Tnx in advance...
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  2. #2
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    Quote Originally Posted by WannaBe View Post
    Let Cn be a cyclic group of order n.
    A. How many sub-groups of order 4 there are in C2xC4... explain.


    Put C_2:=<a>\,,\,\,C_4:=<b>, then it's easy to check that the following are sbgps. of order 4 of C_2\times C_4 :

    \{1\}\times C_4=\{(1,b),(1,b^2),(1,b^3),(1,1)\}\,,\,\,<(a,b)>=  \{(a,b),(1,b^2),(a,b^3),(1,1)\} , <a>\times<b^2>=\{(1,1),(1,b^2),(a,1),(a,b^2)\}

    Try now to mimick the above with the group below.
    B. How many sub-groups of order p there are in CpxCpxC(p^2) when p is a prime? explain.
    C. Prove that if H is cyclic of order 8 then Aut(H) is a non-cyclic group. WHAT is its order?


    If H=<h> , then any \phi \in Aut(H) is completely and uniquely determined by \phi(h) , and since this must be an AUTOMORPHISM it follows that \phi(h) has to be a generator of the group, so \phi(h)=h^r , with (r,8)=1\Longleftrightarrow\,r=1,3,5,7\Longrightarr  ow |Aut(H)|=4.

    Now check that this can't be a cyclic group, say by finding two different elements (i.e., automorphisms) of order 2...


    D. What is the Automorphism group of an infinite cyclic group?

    Just as above: the image of a generator MUST be a generator, so....what are the generators of \mathbb{Z} and what're the possibilities of homomorphisms between them...


    About A-> it's obvious (because it has index 2) that a subgroup of order 4 is normal...

    Oh, dear: much easier! EVERY sbgp. of an abelian group is trivially normal...

    Tonio


    But I can't figure out how many subgroups of this form there are...
    About the other parts-I've no idea...

    I'll be delighted to get guidance about all the parts in this question.

    Tnx in advance...
    .
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  3. #3
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    Hey tonio, 10x for you answer.... But:
    About 1: How did you get to this result? How can we prove that there are no more order 4 sbgps of C2xC4...? Is there a non-brute-force way to check it out?

    About 2: an order p sbgrp is cyclic of course...Hence we need to find all the (a,b,c) in CpxCpxCp^2 such as
    lcm(o(a),o(b),o(c)) = p... p is a prime so one of the elements must have order p and the other two order 1 or p... The options are:
    p, p, p & p,1,1& 1,p,1& 1,1,p& p,p,1 etc... and these are the only options...
    AM I right?

    About 3: Completely understandable...TNX

    About 4: Let H be an infinite cyclic group. Hence H is isomorphic to Z. Let H=<h>... Then we can check the Aut(Z)... Each aut. of Z is determined by the image of the generators of Z... The only generators of Z are 1 and -1... SO there are these possibilities: 1->1, -1->-1 = Identitiy aut. or 1-> -1 , -1->1 and that's it... SO there are 2 automorphisms only...Am I right?

    Your verification& help is needed as you can see... espacially in parts 1&2...

    TNX in advance!
    Last edited by WannaBe; December 30th 2009 at 11:27 AM.
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