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I am interested in families of finite groups arising from direct products of other groups. For instance, abelian groups are a simple example of such families (direct products of cyclic groups), but there are more. The family of hamiltonian groups is another such family; a hamiltonian group is a group where all the subgroups are normal, and it can be expressed as the direct product of the quaternion group of order 8, an abelian group of odd order and a group formed only by involutions. Coxeter groups are other examples; they are represented as direct products of irreducible Coxeter groups.

Could you point out to any other such families??

Thanks in advance, and regards, Guillermo

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Why? Gerhard "Ask Me About System Design" Paseman, 2011.01.11 –  Gerhard Paseman Jan 11 '11 at 22:13
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If $F$ is a family of finite groups, and $I$ is the subfamily of those groups in $F$ which are not isomorphic to a direct product of members of $I$, then every element of $F$ is a direct product of elements of $I$. –  Mariano Suárez-Alvarez Jan 11 '11 at 22:14
    
What choo talking about Mariano? Without more conditions on your family F, you can't claim about direct product (or isomorphic to such) representations of even one member of F. Gerhard "Ask Me About System Design" Paseman, 2011.01.11 –  Gerhard Paseman Jan 11 '11 at 22:21
    
Unless you include trivial products? Gerhard Paseman, 2011.01.11 –  Gerhard Paseman Jan 11 '11 at 22:22
    
Hmm. I seem to be substituting F for I. Never mind. Gerhard "See you later, Ms. Litella" Paseman, 2011.01.11 –  Gerhard Paseman Jan 11 '11 at 22:26
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1 Answer

up vote 3 down vote accepted

A finite group is nilpotent iff it is a direct product of $p$-groups.

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This statement includes "direct product of p-groups". But there may be some groups which arise as direct product of groups of composite order. –  RDK Jan 25 '11 at 10:56
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