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Proving Injectivity of Group Homomorphism Given Relations

  1. Nov 5, 2012 #1
    1. The problem statement, all variables and given/known data

    Let [itex]G = \langle x,y \ | \ x^2, y^3, (xy)^3 \rangle[/itex], and [itex]f: G \rightarrow A_4[/itex] the unique homomorphism such that [itex]f(x) = a[/itex], [itex]f(y) = b[/itex], where [itex]a = (12)(34)[/itex] and [itex]b = (123)[/itex]. Prove that [itex]f[/itex] is an isomorphism. You may assume that it is surjective.

    2. Relevant equations

    N/A

    3. The attempt at a solution

    The previous stages of the question proved that the stated homomorphism was in fact unique, and considered the subgroup [itex]H = \langle y \rangle[/itex] along with the set of cosets [itex]A = \{H, xH, yxH, y^2xH\}[/itex], from which I have proven that [itex]gH \in A, \ \forall g \in G[/itex]. I'm not sure what relevance those stages have to the proof of isomorphism though I'd assume there is some relevance since otherwise the question would be a little disjoint; my only thought so far is that it relates to the first isomorphism theorem for groups, since we have a group homomorphism and a set of cosets, but I can't see how to make it work in this example. We did a similar question in lectures involving proving that [itex]D_{2n} \cong \langle x,y \ | \ x^n, y^2, (xy)^2 \rangle[/itex] where injectivity was proven by rewriting elements of the group as words of the form [itex]x^ky^l[/itex] using the given relations, and concluding that [itex]\#G \leq 2n[/itex]. I have tried something similar with this question but am struggling to rewrite elements of [itex]G[/itex] in a similar form (though I'd guess there'd be an irreducible factor of [itex](xy)^m[/itex] in the word somewhere) which is useful.
     
  2. jcsd
  3. Nov 6, 2012 #2
    I hate to bump, but...anyone?
     
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