definition of Sylow p-subgroup of group
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of p-group.
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:
\( G\): \(\in \{\text{ the groups }\}\)
\( p\): \(\in \{\text{ the prime numbers }\}\)
\( B\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(*G_{p, \beta}\): \(\in \{\text{ the maximal p-subgroups of } G\}\), \(\beta \in B\)
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Conditions:
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"maximal p-subgroup" means that it is a p-subgroup and there is no p-subgroup that contains it.
\(Syl_p (G) := \{G_{p, \beta} \vert \beta \in B\}\), 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 \(A\); \(\cup B \in A\), because for each \(g_1, g_2 \in \cup B\), \(g_1 \in C \in B\) and \(g_2 \in D \in B\), but \(C \subseteq D\) or \(D \subseteq C\); supposing without loss of generality that \(C \subseteq D\), \(g_1, g_2 \in D\); \(g_1 g_2 \in D \in \cup B\); \(1 \in C \in \cup B\); for each \(g \in \cup B\), \(g \in C\), \(g^{-1} \in C\); so, \(\cup B\) is a subgroup; for each \(g \in \cup B\), \(g \in C\), and \(g\) has order of a power of \(p\); \(H \subseteq \cup B\); so, \(\cup B\) is a p-subgroup that contains \(H\), which means that \(\cup B \in A\); then, Zorn's lemma says that there is a maximal p-subgroup that contains \(H\).
When \(G\) is finite, \(\vert G \vert = p_1^{n_1} ... p_k^{n_k}\) for some prime numbers, \(p_1 \lt ... \lt p_k\), and some \(n_1, ..., n_k \in \mathbb{N} \setminus \{0\}\). The order of each Sylow \(p_j\)-subgroup of \(G\) is \(p_j^{n_j}\), which is a part of the Sylow theorem.
