For a prime number p is a p- group in group theory is a group in which the order of each element is a power of p. That is, for each element g of the group, there is a natural number n such that g high pn is equal to the neutral element of the group.

A finite group is a p- group if and only if its group order is a prime power.

The Sylow - sets allow detection of p- subgroups of finite groups with combinatorial methods. Particularly important are the maximum p- subgroups Sylowgruppen a finite group.

  • 3.1 Finite Groups
  • 3.2 Examples of infinite p-groups

Definitions and characteristics

  • A subgroup of a group is called sub- group if it is a group.
  • A subgroup of a group called Sylowuntergruppe - or - Sylowgruppe of if it is maximal subgroup of. That is, for each sub-group of the following from that applies. ( It stands for a fixed prime number. )
  • Groups are special Torsionsgruppen (these are groups in which every element has finite order ).

Special p - groups

P finite groups

  • If G is a finite group, then it is accurate then a p- group if its order is (the number of its elements ) itself has a p- power.
  • The center of a finite non-trivial p- group is nontrivial. This shows you the path formula for conjugation.
  • In the special case of a group of order you can even say more: in this case, the group is abelian. It shows that, by proving generally that the factor group of the group modulo the center, only then is a cyclic group if it is the trivial group.
  • Every finite p- group is nilpotent and hence solvable.
  • A non-trivial finite p- group is simple if, so it has only itself and one group as a normal subgroup if it is isomorphic to.
  • P-groups of the same order need not be isomorphic, eg are the cyclic group C4 and the Klein's four group both 2- groups of order 4, but not to each other isomorphic. A p- group must not be abelian, for example, is the dihedral group a non-abelian 2- group.
  • There are up to isomorphism exactly five groups of order. Three of which are abelian.
  • If a finite group G, the group order and is prime to p, then G contains for each number a p- subgroup H of items. For H is a p- Sylow subgroup, then H is a normal subgroup in a p- subgroup of the group order of G.
  • If, in the situation described a p- Sylow subgroup, then holds, where a subset assigns its normalizer.

Elementary abelian group

Any group is called elementary abelian group if each group member ( except the neutral element ) has order p (p prime) and their link is commutative. Elementary abelian groups are so special abelian p-groups. The term is most often used for finite groups.

  • A finite group G is elementary abelian, if a prime p exists such that G is a finite (internal ) direct product of cyclic subgroups of order p.

An arbitrary, even infinite group is abelian if and only elementary if there exists a prime p such that

  • Each of its finitely generated subgroups is a finite (internal ) direct product of cyclic subgroups of order p or
  • Them as a group is isomorphic to a vector space over the residue field is.

A finite direct product can also be "empty " or only a factor. The trivial, one-element group is therefore also elementary abelian and in this respect at any prime. A non-trivial cyclic group is abelian if and only elementary if it is isomorphic to a finite prime field (as an additive group ).

For these representations is obvious:

  • Every subgroup and every factor group of an elementary abelian group is elementary abelian.

Examples and counter-examples

Finite groups

  • The cyclic group is an abelian p- group and even elementary abelian.
  • The direct product of elementary abelian p is a group.
  • The cyclic group is an abelian p- group which is not elementary abelian.
  • The dihedral and the quaternion group is not abelian 2- groups.
  • No p- group and hence not elementary abelian example is the cyclic group, since it contains elements of order 6 and 6 is not a prime power.
  • Similarly, the symmetric group is not a p- group, since it contains elements of order 2 and elements of order 3, and these orders are not powers of the same prime.

Examples of infinite p-groups

  • Consider the set of all rational numbers whose denominator is 1 or a power of the prime p. With the addition of these numbers modulo 1, we obtain an infinite abelian p- group. Any group that is isomorphic to this, ie group. Groups of this type are important in the classification of infinite abelian groups.
  • The group is also isomorphic to the multiplicative group of those complex roots of unity whose order is a p- power.
  • This group is an abelian p- group but not elementary abelian.
  • The rational function field in one variable (as a group with the addition) an infinite elementary abelian 5 - group.

Pictures of P-group