Equivalence of definitions of reducible multiary group
This article gives a proof/explanation of the equivalence of multiple definitions for the term reducible multiary group
View a complete list of pages giving proofs of equivalence of definitions
Recall that a multiary group is a -ary group for some . The following are equivalent for a -ary group with multiplication :
- There exists a group structure on such that for all (possibly repeated) , with the multiplication on the right being as per the group structure. For more on this, see group is n-ary group for all n
- There exists a neutral element for : evaluated at any tuple where of the entries are equal to and the remaining entry is , gives output . This is true regardless of where we place and is also true if .
- Characterization of subgroup of neutral elements of reducible multiary group
- Groups giving same reducible multiary group are isomorphic
(1) implies (2)
This is immediate: we can take to be the neutral element (i.e., the identity element) of as a group.
(2) implies (1)
Given: A set , function making it a -ary group, an element that is neutral for .
To prove: There exists a group structure on with respect to which for all .
Proof: We use the notation to denote written in succession times.
|Step no.||Assertion/construction||Given data used||Previous steps used||Explanation|
|1||for all and all with .||is neutral for , satisfies associativity becaus it gives a -ary group operation||By asociativity, we have . Using that is neutral, the outer of the left side vanishes and the left side becomes . On the right side, using that is neutral, the inside vanishes and we are left with , completing the proof.|
|2||We can define an operation , and this equals evaluated on any tuple with s, one and one , with the appearing to the left of the .||Step (1)||Step-direct|
|3||The operation defined in Step (2) is associative, i.e., for all||satisfies -ary associativity||Step (2)||We write the operation as . We thus get|
|4||The operation defined in Step (2) has identity element||is neutral for||Step (2)||works as an identity element: for all , . Similarly, .|
|5||The operation defined in Step (2) admits two-sided inverses||Steps (2), (3), (4)||We show one-sided inverses from the unique solution condition: the right inverse of is the unique solution to . Similarly, the left inverse of is the unique such that . Now, we use the equality of left and right inverses in monoid to conclude that two-sided inverses exist.|
|6||The operation defined in Step (2) gives a group structure on||Steps (2), (3), (4), (5)||Step-combination direct|
|7||For any elements with , .||is associative||Steps (2), (6)||We prove this by induction on . Assuming true for , write . The left side simplifies to the desired left side. On the right side, the second input simplifies to by inductive hypothesis, so now using the definition of group product, the product simplifies to as desired.|
|8||We are done||Steps (2), (6), (7)||By Steps (2) and (6), we have a group structure on , which setting in Step (7), gives the desired -ary group, completing the proof.|
Caution about non-uniqueness of neutral elements
For a reducible multiary group, there may be many different choices of neutral element. Each choice yields a potentially different binary operation, though the group structures we obtain are isomorphic. For instance, for the -ary group obtained from the cyclic group of order , every element is a neutral element. The different group structures correspond to affine shifts in the original group, i.e., relabeling all elements by adding a group element to them.