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 . ${\displaystyle G}$ ${\displaystyle |G|=p^{r}\cdot m\;\left(r,m\in \mathbb {N} \setminus \lbrace 0\rbrace \right)}$ ${\displaystyle m}$ ${\displaystyle p,}$ ${\displaystyle G}$ ${\displaystyle s\in \lbrace 0,1,\ldots r\rbrace }$ ${\displaystyle p}$ ${\displaystyle H}$ ${\displaystyle p^{s}}$ ${\displaystyle s=r}$ ${\displaystyle H}$ ${\displaystyle p}$ ${\displaystyle s<r}$ ${\displaystyle H}$ ${\displaystyle p}$ ${\displaystyle p^{s+1}}$ ${\displaystyle G}$ ${\displaystyle H<G}$ ${\displaystyle N_{G}(N_{G}(H))=N_{G}(H)}$ ${\displaystyle N_{G}}$

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 ${\displaystyle g^{p}=e}$