Square map is endomorphism iff abelian
This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself
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- Exponent two implies abelian: If the exponent of a group is 2 (i.e., the group is nontrivial and every non-identity element has order two) then the group is abelian. The analogous statement is not true for any other prime number, i.e., there can be a non-abelian group of prime exponent. The standard example for an odd prime is prime-cube order group:U(3,p) of order .
Other power maps
The power map for a fixed integer is termed a universal power map, and if it is also an endomorphism, it is termed a universal power endomorphism and the group is termed a n-abelian group. This statement gives a necessary and sufficient condition for a group where gives an endomorphism. Here are results for other values of .
- n-abelian iff (1-n)-abelian
- The set of for which is -abelian is termed the exponent semigroup of . It is a submonoid of the multiplicative monoid of integers.
- abelian implies n-abelian for all n
- n-abelian implies every nth power and (n-1)th power commute
- n-abelian implies n(n-1)-central
- n-abelian iff abelian (if order is relatively prime to n(n-1))
- nth power map is surjective endomorphism implies (n-1)th power map is endomorphism taking values in the center
- (n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism
- Frattini-in-center odd-order p-group implies p-power map is endomorphism
- Frattini-in-center odd-order p-group implies (mp plus 1)-power map is automorphism
- Characterization of exponent semigroup of a finite p-group
- Alperin's structure theorem for n-abelian groups
|Value of (note that the condition for is the same as the condition for )||Characterization of -abelian groups||Proof||Other related facts|
|2||abelian groups only||2-abelian iff abelian||endomorphism sends more than three-fourths of elements to squares implies abelian|
|-1||abelian groups only||-1-abelian iff abelian|
|3||3-abelian group means: 2-Engel group and derived subgroup has exponent dividing three||Levi's characterization of 3-abelian groups||cube map is surjective endomorphism implies abelian, cube map is endomorphism iff abelian (if order is not a multiple of 3), cube map is endomorphism implies class three|
|-2||same as for 3-abelian||(based on n-abelian iff (1-n)-abelian)|
Related facts for Lie rings
Here are some related facts for Lie rings:
- Multiplication by n map is a derivation iff derived subring has exponent dividing n
- Multiplication by n map is an endomorphism iff derived subring has exponent dividing n(n-1)
Opposite facts for other algebraic structures
|Statement||Algebraic structure||What step of the proof fails?||Comment|
|Square map is endomorphism not implies abelian for loop||loop||The reparenthesization in Step (3) of the proof below, that requires associativity.||In fact, it is possible to have a noncommutative loop of exponent two.|
|Square map is endomorphism not implies abelian for monoid||monoid||The cancellation in Step (4), which requires that we are working over a cancellative monoid.|
- Associative implies generalized associative: Basically this says that in a group, we can drop and rearrange parentheses at will.
- Invertible implies cancellative in monoid. Since every element of a group is invertible, cancellation is valid in groups.
- Abelian implies universal power map is endomorphism
From square map being endomorphism to abelian
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Given: A group such that the map is an endomorphism, i.e., for all .
To prove: for all .
Proof: We let be arbitrary elements of .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation||What algebraic assumptions does this use?|
|1||--||square map is endomorphism||--||--||None, works over any magma|
|2||--||Step (1)||--||None, just using definition of square. Works over any magma.|
|3||Fact (1)||Step (2)||Reparenthesize||The reparenthesization requires associativity of expressions involving two variables. It works over any semigroup or monoid and even more generally over any diassociative magma.|
|4||Fact (2)||Step (3)||Cancel the right-most from both sides, then the left-most from both sides.||The cancellation requires that we are working in a cancellative magma, such as a cancellative monoid or a quasigroup or loop.|
From abelian to square map being endomorphism
This follows directly from fact (3).