Can there be a p-Sylow subgroup of a p-group

abstract-algebragroup-theorysylow-theory

Have two particular questions about the definition of p-Sylow subgroups:

  1. In all the definitions I've seen, p-Sylow subgroups of a group $G$ are defined as maximal subgroups of $G$ such that their order is a power of p. After a definition makes that point, something like "i.e. a p-subgroup that is not contained in any other p-subgroup". So does "maximal" mean that a given p-Sylow subgroup is not contained in any proper subgroup of $G$, or that that a given p-Sylow subgroup may or may not be contained in a subgroup of $G$ if that subgroup is not a p-group (so maximal only with respect to the $p-$ part.
  2. The main question: Suppose $G$ is a p-group. Then if $H$ is a subgroup of $G$, by one of the definitions above (the latter one), it cannot be a p-Sylow subgroup of $G$ since then it would be contained in a (in this case, not proper, but nowhere is the exclusion made, as far as I know) p-subgroup of $G$ ($G$ itself).

This info has direct impact on a problem I'm working on. So at some point in that problem, we essentially consider a group $G$ and two subgroups $H$ and $K$ with $K$ defined as a p-Sylow subgroup. We eventually prove that $K$ is a subgroup of $H$ and that $H$ is a p-group. If the first definition in my first point is true and $K$ is maximal in the sense that it isn't contained in any proper subgroup of $G$, we need to consider both the case $H$ = $K$ and $H$ = $G$. But if we define it by the later definition and that definition doesn't restrict the 'maximality' of $K$ to proper subgroups of $G$, we need only consider the case $H = K$, because if $G = H$ but $H \neq K$ then that case immediately fails because $H$ not being a proper subgroup of $G$ would not free it from the requirement not to be a p-group.

Best Answer

  1. Maximal means that a $p$-Sylow is not (properly) contained in any other $p$-subgroup of $G$; in other words, it means maximal in the poset of $p$-subgroups. Other subgroups are irrelevant.

  2. Every $p$-group is its own $p$-Sylow. The keyword "other" is important here; it actually implies proper containment if you squint at it, although this is a bit confusing and it would be better to say "properly contained" to be clearer.

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