Endomorphism sends more than three-fourths of elements to squares implies abelian
This article is about a result whose hypothesis or conclusion has to do with the fraction of group elements or tuples of group elements satisfying a particular condition.The fraction involved in this case is 3/4
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- Automorphism sends more than three-fourths of elements to inverses implies abelian
- Inverse map is automorphism iff abelian
- Square map is endomorphism iff abelian
- Cube map is automorphism implies abelian
- The set of elements that commute with any fixed element of the group, is a subgroup: the so-called centralizer of the element.
- The set of elements that commute with every element of the group, is a subgroup: the so-called center of the group
- Subgroup of size more than half is whole group
- Abelian implies universal power map is endomorphism
- The image of a generating set under a homomorphism completely determines the homomorphism.
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Given: A finite group , an endomorphism of . is the subset of comprising those for which . We are given that .
To prove: is abelian and , i.e., for all .
Proof: We initially focus on a single element and try to show that its centralizer is the whole of . We then shift focus to as a set, show that it is contained in the center, and since the center is a subgroup, it is forced to be all of .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation||Commentary|
|1||Suppose are such that are all in . Then||--||Definition of as the set of elements mapped to their squares, is an endomorphism||[SHOW MORE]||Convert local property of elements going to squares to local property of pairs commuting.|
|2||For any , define . We have , where is the centralizer of in .||Step (1)||[SHOW MORE]||Toward establishing a large centralizer for fixed but arbitrary element of -- use local property of elements going to squares.|
|3||For any , the intersection has size greater than||[SHOW MORE]||Toward establishing a large centralizer for fixed but arbitrary element of -- use size constraints and combinatorial formula.|
|4||For any , the subgroup is equal to the whole group||Facts (1), (3)||Steps (2), (3)||[SHOW MORE]||Complete establishing that a fixed but arbitrary element of has centralizer the whole group -- use that big enough subgroups must contain everything.|
|5||is contained in the center of||Step (4)||[SHOW MORE]||Shift focus to as a set.|
|6||equals its own center, hence is abelian||Facts (2), (3)||Step (5)||[SHOW MORE]||Clinch -- use that big enough subgroups must contain everything.|
|7||is a generating set for .||Fact (3)||--||[SHOW MORE]|
|8||, i.e., for all .||Facts (4), (5)||--||Steps (6), (7)||[SHOW MORE]|
Steps (6) and (8) clinch the proof.