Sylow $p$-subgroups of a finite group

Depends on

• A finite $p$-group has order $p^n$ for a prime $p$ and some $n\in\mathbb N$
• The $p$-adic valuation $v_p(a)$ of a nonzero integer: the greatest $k \in \mathbb{N}$ with $p^{k} \mid a$

Used by

• Frobenius automizer criterion for p nilpotence Corollary
• Cyclic sylow does not alone imply a normal p complement Counterexample
• Control of fusion in a sylow p subgroup Definition
• Normal p complement and p nilpotent group Definition
• P local normalizer for normal complement theory Definition
• p-complements in finite groups Definition
• The number nₚ(G) of Sylow p-subgroups Definition
• Frobenius normal two complement for S₃ Example
• Sylow p-subgroups of Aut((ℤ/p)²): nₚ=p+1 Example
• Abelian sylow fusion in its normalizer Lemma
• Distinct normal Sylow subgroups centralize one another Lemma
• Elementary detection at a fixed element Lemma
• Fusion control and centralizer transitivity are equivalent Lemma
• Fusion control forces trivial Sylow intersection with the p residual Lemma
• Local normal p complements force control of fusion Lemma
• Local sylow conjugacy ascent for fusion Lemma
• P automizer condition implies fusion control Lemma
• P residual is generated by p prime elements and idempotent Lemma
• Sylow subgroups of a normal subgroup are intersections with Sylow subgroups Lemma
• Sylow times normal subgroup covers when the index is a p-power Lemma
• The local automizer condition gives centralizer conjugacy of Sylow subgroups Lemma
• Equivalent forms of having a normal p complement Proposition
• Burnside normal p complement theorem Theorem
• Frobenius normal p complement theorem Theorem
• Sylow I: every finite group has a Sylow p-subgroup Theorem

Dependency tree · two levels

26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Keith Conrad, Consequences of the Sylow Theorems, Sections 1-5 (standard reference, not scraped)