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Showing a group is infinite and nonabelian given its presentation

  1. Oct 14, 2012 #1
    1. The problem statement, all variables and given/known data

    The question is out of Hungerford's Algebra (Graduate Texts in Mathematics). Page 69,#7:

    Show that the group defined by generators a,b and relations [itex]a^2=e, b^3 = e[/itex] is infinite and nonabelian.


    2. Relevant equations



    3. The attempt at a solution

    My professor gave hints, suggesting that we construct an onto homomorphism using Van Dyck's Theorem. Here is a sketch of my proof:

    Let [itex] G = \langle a, b | a^2 = e, b^3 = e \rangle[/itex]. By Van Dyck's Theorem, there exists a unique onto homomorphism from G to [itex]D_3[/itex]. Note that [itex] D_3 = \langle a^i b^j : 0 \leq i \leq 1, 0 \leq j \leq 2 \rangle[/itex]. Thus G is nonabelian since [itex]D_3[/itex] is nonabelian.

    To show that G is infinite consider [itex] \alpha, \beta \in S_\mathbb{N} [/itex], where α = (34)(67)... and β = (123)(456)... . Here o(α) = 2 and o(β) = 3, but [itex] |\langle \alpha, \beta \rangle | = \infty [/itex]. Again, by Van Dyck's Theorem, there exists a unique onto homomorphism from G to [itex] \langle\alpha, \beta \rangle [/itex]. Therefore G is infinite. [itex] \blacksquare[/itex]



    Why does the existence and uniqueness of an onto homomorphism to a group with these properties give the group G the desired properties?
     
  2. jcsd
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