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I know this should be obvious but somehow I can't seem to figure it out and it annoys me! My definition of nilpotent groups is the following: A group $G$ is nilpotent if every subgroup of $G$ is subnormal in $G$, or equivalently if $U for all $U. And my definition of solvable groups is that a group $G$ is solvable if $U'\neq U$ for all subgroups $1\neq U \le G$, where $U'$ is the commutator subgroup of $U$. My question is then: why are nilpotent groups solvable?

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    You've marked this as "finite groups". Are you concerned *solely* with finite groups? Do you know that finite nilpotent groups are products of $p$-groups?2012-05-12
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    @ArturoMagidin: The given definition for *nilpotent* works only for finite groups (same for *solvable*).2012-05-13
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    @jug: Right; just like "product of $p$-groups" only works for finite groups. It should still be stated explicitly rather than implicitly, don't you think?2012-05-13

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