# 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. ${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle g}$ ${\ displaystyle n}$${\ displaystyle g}$ ${\ displaystyle p ^ {n}}$

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 . ${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle p}$

## Definitions and characteristics

• A subgroup of a group is called a subgroup if it is a group.${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle p}$${\ displaystyle p}$
• A subgroup of a group is called a -Sylow subgroup or -Sylow group of if it is the maximum subgroup of . That is, for every subgroup of it follows that . (Here stands for a fixed prime number.)${\ displaystyle p}$${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle G}$${\ displaystyle p}$${\ displaystyle G}$${\ displaystyle p}$${\ displaystyle U}$${\ displaystyle G}$${\ displaystyle H \ subseteq U}$${\ displaystyle H = U}$${\ displaystyle p}$
• ${\ displaystyle p}$-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 .${\ displaystyle G}$${\ displaystyle p}$${\ displaystyle p}$
• The center of a finite nontrivial group is itself a nontrivial group. This is shown with the orbit formula for conjugation .${\ displaystyle p}$${\ displaystyle p}$
• In the special case of a group of the order one can say even more: In this case the group is either isomorphic to the cyclic group or to the direct product . In particular, the group is Abelian.${\ displaystyle p ^ {2}}$${\ displaystyle C_ {p ^ {2}}}$${\ displaystyle C_ {p} \ times C_ {p}}$
• Every finite group is nilpotent and therefore also solvable .${\ displaystyle p}$
• A nontrivial finite group is then exactly easy , so it has only the trivial normal subgroup if it has elements and isomorphic with it is.${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle C_ {p}}$
• ${\ displaystyle p}$-Groups of the same order need not be isomorphic , e.g. B. the cyclic group and the Klein group of four are both 2-groups of order 4, but not isomorphic to one another. A group does not have to be Abelian either , e.g. B. the dihedral group is a nonabelian 2 group.${\ displaystyle C_ {4}}$${\ displaystyle p}$ ${\ displaystyle D_ {8}}$
• Except for isomorphism, there are exactly five groups of the order . Three of them are Abelian.${\ displaystyle p ^ {3}}$
• Except for isomorphism, there are exactly P (n) abelian groups of the order . Where P is the partition function .${\ displaystyle p ^ {n}}$
• 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 .${\ displaystyle G}$${\ displaystyle | G | = p ^ {r} \ cdot m \; \ left (r, m \ in \ mathbb {N} \ setminus \ lbrace 0 \ rbrace \ right)}$${\ displaystyle m}$${\ displaystyle p,}$${\ displaystyle G}$${\ displaystyle s \ in \ lbrace 0,1, \ ldots r \ rbrace}$${\ displaystyle p}$${\ displaystyle H}$${\ displaystyle p ^ {s}}$${\ displaystyle s = r}$${\ displaystyle H}$${\ displaystyle p}$${\ displaystyle s ${\ displaystyle H}$${\ displaystyle p}$${\ displaystyle p ^ {s + 1}}$${\ displaystyle G}$${\ displaystyle H ${\ displaystyle N_ {G} (N_ {G} (H)) = N_ {G} (H)}$${\ displaystyle N_ {G}}$

### 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 .${\ displaystyle \ mathbb {Z} / p \ mathbb {Z}}$ ${\ displaystyle (V, +)}$ ${\ displaystyle \ mathbb {Z} / p \ mathbb {Z}}$

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.${\ displaystyle C_ {p}}$
• The direct product is an elementary abelian p group.${\ displaystyle C_ {p} \ times C_ {p}}$
• The cyclic group is an abelian p group that is not elementary abelian.${\ displaystyle C_ {p ^ {2}}}$
• The dihedral group and the quaternion group are not Abelian 2 groups.${\ displaystyle D_ {8}}$ ${\ displaystyle Q_ {8}}$
• 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.${\ displaystyle C_ {6} \ cong C_ {2} \ times C_ {3}}$
• 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.${\ displaystyle S_ {3}}$

### 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.${\ displaystyle p ^ {\ infty}}$
• 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.${\ displaystyle p ^ {\ infty}}$
• The rational function body in a variable is (as a group with addition) an infinite elementary Abelian 5 group.${\ displaystyle \ mathbb {Z} / 5 \ mathbb {Z} (t)}$

## 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

1. In this article always stands for a prime number${\ displaystyle p}$
2. Hungerford p. 93
3. Hungerford p. 94
4. Hungerford 7.1
5. ^ Hungerford p. 95, this is a tightening of the 1st Sylow theorem.
6. Hungerford also counts this combinatorial conclusion from the orbit formula to the Sylow theorems.
7. For finite groups the commutativity follows from the first requirement that all elements satisfy, for infinite groups it is additionally required. See Hungerford${\ displaystyle g ^ {p} = e}$