Group of finite exponent: Difference between revisions
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==Definition== | ==Definition== | ||
A '''group of finite exponent''' is a [[group]] | A '''group of finite exponent''' is a [[group]] satisfying the following equivalent conditions: | ||
# Its [[exponent of a group|exponent]] is a finite nonnegative integer. In other words, all the elements of the group have finite [[order of an element|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 nonnegative integer. In other words, all the elements of the group have finite [[order of an element|order]], and the maximum of the orders of all elements is finite. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::finite group]] || || || || {{intermediate notions short|group of finite exponent|finite group}} | |||
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===Weaker properties=== | ===Weaker properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Stronger than::periodic group]] || all elements have finite order, but there need not be a uniform bound on the orders of elements. || || || || {{intermediate notions short|periodic group|group of finite exponent}} | |||
Revision as of 15:10, 17 April 2013
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
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 nonnegative integer. 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 nonnegative integer. 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 |