2024-11-25

872: Sylow p-Subgroup of Group

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definition of Sylow p-subgroup of group

Topics


About: group

The table of contents of this article

Starting Context

Target Context


  • The reader will have a definition of Sylow p-subgroup of group.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

Main Body

1: Structured Description


Here is the rules of Structured Description.

Entities:
G: ∈{ the groups }
p: ∈{ the prime numbers }
B: ∈{ the possibly uncountable index sets }
∗Gp,β: ∈{ the maximal p-subgroups of G}, β∈B
//

Conditions:
//

"maximal p-subgroup" means that it is a p-subgroup and there is no p-subgroup that contains it.

Sylp(G):={Gp,β|β∈B}, which is the set of all the Sylow p-subgroups of G.

2: Note


For an arbitrary p, there may be no Sylow p-subgroup, because there may be no p-subgroup.

If there is a p-subgroup, H, there will be a Sylow p-subgroup that contains H, which is by Zorn's lemma: let A be the set of the p-subgroups that contains H; let B be any nonempty chain of A; ∪B∈A, because for each g1,g2∈∪B, g1∈C∈B and g2∈D∈B, but C⊆D or D⊆C; supposing without loss of generality that C⊆D, g1,g2∈D; g1g2∈D∈∪B; 1∈C∈∪B; for each g∈∪B, g∈C, g−1∈C; so, ∪B is a subgroup; for each g∈∪B, g∈C, and g has order of a power of p; H⊆∪B; so, ∪B is a p-subgroup that contains H, which means that ∪B∈A; then, Zorn's lemma says that there is a maximal p-subgroup that contains H.

When G is finite, |G|=p1n1...pknk for some prime numbers, p1<...<pk, and some n1,...,nk∈N∖{0}. The order of each Sylow pj-subgroup of G is pjnj, which is a part of the Sylow theorem.

References
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