# p group

For a prime number , a **group** in group theory is a group in which the order of each element is a power of . That is, there is a natural number for each element of the group , so that high equals the neutral element of the group.
** **

The Sylow theorems make it possible to find subsets of finite groups with combinatorial methods. The maximum subgroups, the **Sylow groups of** a finite group, are particularly important .

## Definitions and characteristics

- A subgroup of a group is called a subgroup if it is a group.
- A subgroup of a group is called a -Sylow subgroup or -Sylow group of if it is the maximum subgroup of . That is, for every subgroup of it follows that . (Here stands for a fixed prime number.)

- -Groups are special torsion groups (these are groups in which each element has finite order).

## Special *p* groups

### Finite *p* groups

- If a finite group, then it is a -group if and only if its order is a power of .
- The center of a finite nontrivial group is itself a nontrivial group. This is shown with the orbit formula for conjugation .
- In the special case of a group of the order one can say even more: In this case the group is either isomorphic to the cyclic group or to the direct product . In particular, the group is Abelian.
- Every finite group is nilpotent and therefore also solvable .
- A nontrivial finite group is then exactly easy , so it has only the trivial normal subgroup if it has elements and isomorphic with it is.
- -Groups of the same order need not be isomorphic , e.g. B. the cyclic group and the Klein group of four are both 2-groups of order 4, but not isomorphic to one another. A group does not have to be Abelian either , e.g. B. the dihedral group is a nonabelian 2 group.
- Except for isomorphism, there are exactly five groups of the order . Three of them are Abelian.
- Except for isomorphism, there are exactly P (n) abelian groups of the order . Where P is the partition function .
- If a finite group has the group order and is coprime to then contains a subgroup with elements for every number . For is a -Sylow subgroup. If is, then a normal subgroup is in a subgroup with the group order of . If a
*p*-Sylow subgroup is in the situation described , then the following applies , where a subgroup is assigned its normalizer .

### Elementary Abelian group

An arbitrary group is called an **elementary Abelian group** if every group element (except for the neutral element) has order *p* ( *p* prime) and its combination is commutative . Elementary Abelian groups are therefore special Abelian *p* groups. The term is mostly used for finite groups.

- A finite group
*G*is elementary Abelian if and only if there is a prime*p*such that*G is*a finite (inner) direct product of cyclic subgroups of order*p*.

Any group, including an infinite group, is elementary Abelian if and only if there is a prime *p* such that

- each of its finitely producible subgroups is a finite (inner) direct product of cyclic subgroups of order
*p*or - as a group it is isomorphic to a - vector space over the remainder class field .

A finite direct product can also be “empty” here or only have one factor. The trivial, one-element group is also elementary Abelian and this with respect to every prime number. A nontrivial cyclic group is elementary Abelian if and only if it is isomorphic to a finite prime field (as an additive group).

From the above representations it becomes obvious:

- Every subgroup and every factor group of an elementary Abelian group is elementary Abelian.

## Examples and counterexamples

### Finite groups

- The cyclic group is an abelian
*p*group and is even elementary abelian. - The direct product is an elementary abelian
*p*group. - The cyclic group is an abelian
*p*group that is not elementary abelian. - The dihedral group and the quaternion group are not Abelian 2 groups.
- No
*p*group and therefore not elementary Abelian is z. B. the cyclic group , since it contains elements of order 6 and 6 is not a prime power. - Likewise, 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 number
*p*. Adding these numbers modulo 1 gives us an infinite Abelian*p*group. Any group that is isomorphic to this is called a group. Groups of this type are important in classifying infinite Abelian groups.

- The group is also isomorphic to the multiplicative group of those complex roots of unity , the order of which is a
*p*power. This group is an abelian*p*group but not elementary abelian.

- The group is also isomorphic to the multiplicative group of those complex roots of unity , the order of which is a

- The rational function body in a variable is (as a group with addition) an infinite elementary Abelian 5 group.

## literature

- Thomas W. Hungerford:
*Algebra*(=*Graduate Texts in Mathematics.*Vol. 73). 5th printing. Springer, New York NY et al. 1989, ISBN 0-387-90518-9 , Chapter I Groups, 5-7.

## Individual evidence

- In this article ↑ always stands for a prime number
- ↑ Hungerford p. 93
- ↑ Hungerford p. 94
- ↑ Hungerford 7.1
- ^ Hungerford p. 95, this is a tightening of the 1st Sylow theorem.
- ↑ Hungerford also counts this combinatorial conclusion from the orbit formula to the Sylow theorems.
- ↑ For finite groups the commutativity follows from the first requirement that all elements satisfy, for infinite groups it is additionally required. See Hungerford