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$\begingroup$

Every group of order $231$ is the direct product of a group of order $11$ and a group of order $21$.

By Sylow's theorem, we know there are one Sylow-7 subgroup, one Sylow-11 subgroup (these two are normal, for sure), and some Sylow-2 subgroups.

What does the "direct product" mean in this context? Thanks.

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    http://en.wikipedia.org/wiki/Direct_product_of_groups2012-10-06
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    @QiaochuYuan In that wiki page, $|G\times H|=|G||H|$, which seems not true in this context. :)2012-10-07
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    I don't follow. The last time I checked, $231 = 11 \cdot 21$.2012-10-07
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    @QiaochuYuan Hmmm... I was writing it as a "direct product" of a Sylow-7, a Sylow-11 and some Sylow-3 subgroups, whose intersection with each other are all trivial. So in fact, this is not a _direct_ _product_... Could you show me how to deal with it?2012-10-07
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    The Sylow $3$-subgroup and the Sylow $7$-subgroup need not commute with each other in general.2012-10-07

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