finitely generated groups

**mms** · July 17th 2009, 03:24 PM

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!

**NonCommAlg** · July 17th 2009, 04:09 PM

Originally Posted by mms

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!

let $H=<h_1, \cdots , h_m>, \ G/H = <Hg_1, \cdots , Hg_n>,$ and $g \in G.$ then $g \in H<g_1, \cdots , g_n> \subseteq <h_1, \cdots ,h_m,g_1, \cdots , g_n>.$ thus: $G \subseteq <h_1, \cdots , h_m, g_1, \cdots , g_n> \subseteq G.$

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