Abelian group

Abelian group
Tag	`071DDFVG4J`
URI	`m2://071DDFVG4J`
Kind	definition
Writing	W1 stated
MSC	20Kxx
Defines	Abelian group
Uses	Group, Commutativity

Markdown · JSON · Edit

Definition

An abelian group is a group $(G, \ast)$ whose operation is commutative. Thus $a \ast b = b \ast a$ for all $a, b \in G$ .

Remarks

The integers, the rationals and the real numbers with addition are abelian groups. The operation of an abelian group is often written $+$ , with identity $0$ and inverse $-a$ . The group of invertible $2 \times 2$ real matrices under multiplication is not abelian.

Explanations

L1 human

In an abelian group, the order of two elements in a product does not matter. Addition of numbers is the model: $a + b = b + a$ . The name honours Niels Henrik Abel.

What this node needs for W2

edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 4 nodes in this order. The foundations come first, and the list ends with this node.

Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.

no evidence level

Binary operation definition
Group concept
Commutativity definition
Abelian group definition

What depends on this

22 nodes use this node, directly or through other nodes.

Extensions of solvable groups are solvable 01EY0EM7RP
Quotients of solvable groups are solvable 027ZQS04SJ
Infinite cyclic group 02B1T50JD1
Circle group 04EAFVFTF7
Cyclic groups are abelian 04RG02S6BA
Subgroups of solvable groups are solvable 05Y0F9M0TS
Classification of finite abelian groups 07CQR9BW3R
Additive group of integers 0BSCJA5G53
Burnside's p^a q^b theorem 0FMAQSB1SC
Nilpotent groups are solvable 0GNG2KVPZJ
The symmetric group on five or more letters is not solvable 0J53NE674D
Subgroups of abelian groups are normal 0JPDTEDCFD
A finite abelian group is cyclic iff its exponent equals its order 0MC0AQZSHE
Solvable group 0P3KZK0FTV
Klein four-group 0PMRQJVTME
Abelian groups are solvable 0R0X73XHAT
Torsion subgroup 0RH1TPDGCQ
Structure theorem for finitely generated abelian groups 0RY0B72XAS
Subgroups of the integers 0SRATCFF1W
Group of nth roots of unity 0WX11EA3F3
Integers modulo n as a quotient group 0YJ3WRR9RY
Free abelian group 0YMSEZE1VH

Sources

Web https://en.wikipedia.org/wiki/Abelian_group, Definition

What links here

Used by

Circle group 04EAFVFTF7
Cyclic groups are abelian 04RG02S6BA
Classification of finite abelian groups 07CQR9BW3R
Additive group of integers 0BSCJA5G53
Subgroups of abelian groups are normal 0JPDTEDCFD
A finite abelian group is cyclic iff its exponent equals its order 0MC0AQZSHE
Solvable group 0P3KZK0FTV
Klein four-group 0PMRQJVTME
Abelian groups are solvable 0R0X73XHAT
Torsion subgroup 0RH1TPDGCQ
Structure theorem for finitely generated abelian groups 0RY0B72XAS
Group of nth roots of unity 0WX11EA3F3
Free abelian group 0YMSEZE1VH