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$p$-subgroup (Definition)

Let $G$ be a finite group with order $n$, and let $p$ be a prime integer. We can write $n=p^k m$ for some $k,m$ integers, such that $k$ and $m$ are coprimes (that is, $p^k$ is the highest power of $p$ that divides $n$). Any subgroup of $G$ whose order is $p^k$ is called a Sylow $p$-subgroup.

While there is no reason for Sylow $p$-subgroups to exist for any finite group, the fact is that all groups have Sylow $p$-subgroups for every prime $p$ that divides $\vert G\vert$. This statement is the First Sylow theorem

When $\vert G\vert=p^k$ we simply say that $G$ is a $p$-group.



"$p$-subgroup" is owned by drini. [ full author list (3) | owner history (2) ]
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See Also: $p$-group

Also defines:  Sylow $p$-subgroup, Sylow p-subgroup, $p$-group, p-group
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Cross-references: Sylow theorem, groups, subgroup, divides, power, coprimes, integers, prime integer, order, finite group
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This is version 5 of $p$-subgroup, born on 2003-10-15, modified 2004-12-12.
Object id is 5184, canonical name is PGroup.
Accessed 4379 times total.

Classification:
AMS MSC: 20D20 (Group theory and generalizations :: Abstract finite groups :: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure)
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