Group of finite exponent
From Groupprops
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Definition
A group of finite exponent is a group satisfying the following equivalent conditions:
- Its exponent is a finite natural number. In other words, all the elements of the group have finite order, and the lcm of the orders of all elements (which is how the exponent is defined) is finite.
- The maximum of element orders is a finite natural number. In other words, all the elements of the group have finite order, and the maximum of the orders of all elements is finite.
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
finite group | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions | |
---|---|---|---|---|---|
periodic group | all elements have finite order, but there need not be a uniform bound on the orders of elements. | |FULL LIST, MORE INFO |