Sylow sentences

The Sylow theorems (after Ludwig Sylow ) are three mathematical theorems from group theory , a branch of algebra . They make it possible to make statements about subgroups of finite groups and also to classify some groups of finite order .

In contrast to finite cyclic groups , one can generally not say anything about the existence and number of subgroups for any finite groups. We only know from Lagrange's theorem that a subgroup of a group has an order that is a divisor of the order of . The Sylow theorems provide additional statements here, but also do not allow a complete classification of finite groups. This takes place via the classification of the finite simple groups . ${\displaystyle G}$ ${\displaystyle G}$

In addition to Sylow (1872), Eugen Netto and Alfredo Capelli , among others, gave evidence.

The sentences

In the following, be a finite group of the order , where a prime number and a natural number that is too prime . A maximal subgroup of is called a Sylow subgroup . ${\displaystyle G}$ ${\displaystyle |G|=p^{r}m}$ ${\displaystyle p}$ ${\displaystyle m}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle p}$

1. For everyone possesses a subgroup of order . In particular, the maximal -subgroups of have the order . ${\displaystyle 0\leq s\leq r}$ ${\displaystyle G}$ ${\displaystyle p^{s}}$ ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle p^{r}}$
2. Be a -Sylow subgroup. Then of each subgroup that is p-group contains one conjugate. So there is one with . ${\displaystyle P<G}$ ${\displaystyle p}$ ${\displaystyle P}$ ${\displaystyle U<G}$ ${\displaystyle g\in G}$ ${\displaystyle gUg^{-1}\subseteq P}$
3. The number of -Sylow subgroups is a factor of the index of the -Sylow subgroup of and of the form with . ${\displaystyle p}$ ${\displaystyle m}$ ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle 1+kp}$ ${\displaystyle k\in \mathbb {N} _{0}}$

Inferences

• Cauchy's theorem : If there is a group whose order is shared by a prime number , there is an element of order . ${\displaystyle G}$ ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle p}$
• Every two -Sylow groups of a group are conjugated and therefore isomorphic. ${\displaystyle p}$ ${\displaystyle G}$
• Be a group and a -Sylow subgroup. Then is the normal subgroup of if and only if the only -Sylow subgroup is of. ${\displaystyle G}$ ${\displaystyle P<G}$ ${\displaystyle p}$ ${\displaystyle P}$ ${\displaystyle G}$ ${\displaystyle P}$ ${\displaystyle p}$ ${\displaystyle G}$
• Let be a finite group whose order is shared by a prime number . If Abelian, there is only one -Sylow subgroup in . ${\displaystyle G}$ ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle p}$ ${\displaystyle G}$

Examples

Each group of order 15 is cyclic

Be a group of order . If one denotes with the number of 3-Sylow subgroups of and with the number of 5-Sylow subgroups of , the following applies: ${\displaystyle G}$ ${\displaystyle |G|=15=3\cdot 5}$ ${\displaystyle s_{3}}$ ${\displaystyle G}$ ${\displaystyle s_{5}}$ ${\displaystyle G}$

1. ${\displaystyle s_{3}\equiv 1\;\operatorname {mod} \;3}$and , therefore, must apply. ${\displaystyle s_{3}\mid 5}$ ${\displaystyle s_{3}=1}$
2. ${\displaystyle s_{5}\equiv 1\;\operatorname {mod} \;5}$and , therefore, must apply. ${\displaystyle s_{5}\mid 3}$ ${\displaystyle s_{5}=1}$

So the 3-Sylow subgroup and the 5-Sylow subgroup are normal divisors of G. As p -subgroups to different prime numbers, their average is , where the neutral element denotes. Therefore their complex product is direct, that is (see complementary normal divisors and direct product ). Since the direct product has order 15, it must be, and follows with the Chinese remainder . ${\displaystyle G_{3}}$ ${\displaystyle G_{5}}$ ${\displaystyle \{e\}}$ ${\displaystyle e\in G}$ ${\displaystyle G}$ ${\displaystyle G_{3}\cdot G_{5}\simeq G_{3}\times G_{5}}$ ${\displaystyle G\simeq \mathbb {Z} /3\mathbb {Z} \times \mathbb {Z} /5\mathbb {Z} }$ ${\displaystyle G\simeq \mathbb {Z} /15\mathbb {Z} }$

There is no simple group of order 162

Be . After the Sylow theorems there is a subgroup of the order (namely a 3-Sylow group). This is from index 2, so normal. is therefore not easy. ${\displaystyle |G|=162=2\cdot 3^{4}}$ ${\displaystyle 3^{4}=81}$ ${\displaystyle G}$

literature

• Ludwig Sylow: Théorèmes sur les groupes de substitutions . In: Mathematische Annalen , Volume 5, 1872, pp. 584-594
• Kurt Meyberg: Algebra - Part 1 . Hanser, 1980, ISBN 3-446-13079-9 , pp. 69-77
• Michael Holz: Repetition of Algebra . Binomi, 2005, ISBN 3-923923-44-9 , pp. 251-263

Web links

Wikibooks: Proof of Sylow's Theorems  - Learning and Teaching Materials