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I know some facts such as: Nilpotent groups are solvable, $p$-groups are nilpotent, a finite group whose order is a product of distinct primes is solvable, and finite groups are nilpotent if and only if it is a finite product of solvable groups.

Are there any other major relationships between these concepts that one should be aware of?

Also, how do you compute a transfer? Could you provide a simple example or two?

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    Have you heard of the M-groups? If not, I recommend [the book](http://www.amazon.com/Character-Theory-Finite-Groups-Mathematics/dp/0486680142/ref=la_B001HMLSW4_1_3?ie=UTF8&qid=1355314566&sr=1-3)2012-12-12
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    Not every product of solvable groups is nilpotent, for example $S_3 \times \mathbb{Z}_2$ is not nilpotent. However, it is true that a finite group is nilpotent if and only if it is the product of its Sylow subgroups.2012-12-12
  • 0
    You want to read the book "Between nilpotent and solvable" by Henry G. Bray and Michael Weinstein. The whole book is about a big hierarchy of notions that lie between the two in the title. This includes M-groups, equichained groups, and others.2012-12-12
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    It is not true that all $p$-groups are nilpotent.2012-12-12
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    http://math.stackexchange.com/a/205805/12952 This earlier answer of mine may help you with the first part.2012-12-12
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    Probably, he thinks that all groups are finite. For me that's also true ;-)2012-12-12

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