Show that if H is a normal subgroup of a group G such that H and G/H are finitely generated, then so is G

thanks! (Hi)

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- Jul 17th 2009, 03:24 PMmmsfinitely generated groups
Show that if H is a normal subgroup of a group G such that H and G/H are finitely generated, then so is G

thanks! (Hi) - Jul 17th 2009, 04:09 PMNonCommAlg
let $\displaystyle H=<h_1, \cdots , h_m>, \ G/H = <Hg_1, \cdots , Hg_n>,$ and $\displaystyle g \in G.$ then $\displaystyle g \in H<g_1, \cdots , g_n> \subseteq <h_1, \cdots ,h_m,g_1, \cdots , g_n>.$ thus: $\displaystyle G \subseteq <h_1, \cdots , h_m, g_1, \cdots , g_n> \subseteq G.$