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Find a direct summand of a finite abelian group

  1. Jul 20, 2012 #1
    1. The problem statement, all variables and given/known data

    If G is a finite abelian group, and x is an element of maximal order, then <x> is a direct summand of G.

    2. Relevant equations



    3. The attempt at a solution

    I claim that the hypothesis implies that A = G\<x> [itex]\bigcup[/itex] {e} is a subgroup of G. If so, then since G = < <x> [itex]\bigcup[/itex] A>, and <x> [itex]\bigcap[/itex] A = <e>, that G = <x> [itex]\oplus[/itex] A.

    Pf of claim: A is obviously associative, has an identity by definition, and since <x> is a group, b [itex]\in[/itex] A [itex]\Rightarrow[/itex] b^-1 [itex]\in[/itex] A. I'm struggling to show that A is closed.

    I feel the key is that I must show that if a,b [itex]\in[/itex] G, and [itex]\exists[/itex]x such that ab [itex]\in[/itex] <x>, then [itex]\exists[/itex]y[itex]\in[/itex]G such that a,b [itex]\in[/itex] <y>. Then the maximality of |x| will require y=x, whence A is closed. But I need a nudge here. I just can't get a rigorous proof of this.
     
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  3. Jul 20, 2012 #2

    micromass

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    I think that what you're trying to prove is false. That is: [itex](G\setminus <x>) \cup \{e\}[/itex] is not a subgroup.
     
  4. Jul 20, 2012 #3
    Since G is finite abelian, G [itex]\cong[/itex] [itex]\sum[/itex] [itex]\textbf{Z}_{p_{i}^{n_i}}[/itex]. If |x| has maximal order, then isn't A just removing one of the direct summands, so that, say, A [itex]\cong[/itex] [itex]\sum[/itex] [itex]<e> \oplus \textbf{Z}_{p_{i}^{n_i}}[/itex], with [itex] i \geq 2[/itex], which is a subgroup?
     
    Last edited: Jul 20, 2012
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