Definition:Infinite Group
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Definition
A group which is not finite is an infinite group.
That is, an infinite group is a group of infinite order.
That is, a group $\struct {G, \circ}$ is an infinite group if and only if its underlying set $G$ is infinite.
That is, an infinite group is a group with an infinite number of elements.
Countable
An infinite group whose underlying set $G$ is countable is a countably infinite group.
Uncountable
An infinite group whose underlying set is uncountable is an uncountable group.
Also see
- Results about the order of a group can be found here.
- Results about infinite groups can be found here.
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text{I}$: The Group Concept: $\S 3$: Examples of Infinite Groups
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.4$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Problem $\text{GG}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 38$
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Definition $1.2$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): group
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): infinite group
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): group
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): infinite group