Sylow theorems

The Sylow -phrases (after Ludwig Sylow ) are three mathematical theorems from algebra. They make it possible to make statements about subgroups of finite groups and classify some groups of finite order.

In contrast to finite cyclic groups one can say nothing about the existence and number of subsets in any finite groups in general. It is known only from the set of Lagrange, that a subgroup of a group has an order which divisor of the order of is. The Sylowsätze here provide additional statements, but also do not allow complete classification of finite groups. This takes place on the classification of finite simple groups.

Besides Sylow (1872 ) stated, among other things Eugen Netto and Alfredo Capelli evidence.

  • 2.1 Each group of order 15 is cyclic
  • 2.2 There is no simple group of order 162

The sentences

Be in the following a finite group of order, which is a prime number, and a natural number relatively prime to be.

Conclusions

  • If a group whose order is divided by a prime number, so there is an element of order.
  • The two - Sylowgruppen a group are conjugate (and hence isomorphic ).
  • Let be a group and one - Sylowgruppe. The following applies:
  • Let be a finite group whose order is divided by a prime number. Is abelian, then there is only one Sylow p-group in.

Examples

Each group of order 15 is cyclic

Let be a group of order. If we denote by the number of 3- Sylowuntergruppen by and with the number of 5 - of Sylowuntergruppen, then:

So the 3- Sylowuntergruppe and the 5- Sylowuntergruppe normal subgroup of G. As are p- subgroups at different primes intersect at, wherein the neutral element of designated. Hence their complex product is direct. (see Complementary normal subgroups and direct product ). Since the direct product has order 15, it follows with the Chinese remainder theorem

There is no simple group of order 162

Be.

Off and follows

So the 3- Sylowgruppe is a normal subgroup of order. This normal subgroups can thus be neither the whole group, nor can it consist only of the identity element. is therefore not easy.

Lagrange's theorem (group theory)
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