# Groups giving same reducible multiary group are isomorphic

## Contents

## Statement

Suppose is a -ary group (i.e., a multiary group with arity ) with -ary operation that is reducible. Suppose and are multiplications on , both making into a group, that both induce the -ary operation when thought of the usual way:

for all

Then, the groups and are isomorphic groups, and the isomorphism can be expressed explicitly in either group in terms of multiplication by a central element of order dividing .

## Related facts

- Equivalence of definitions of reducible multiary group
- Characterization of subgroup of neutral elements of reducible multiary group

## Facts used

## Proof

**Given**: Set with two group operations and such that:

for all

**To prove**: and are isomorphic groups

**Proof**: For simplicity, we will denote by concatenation but write explicitly. Let be the identity element for and be the identity element for .
is a neutral element for , hence by Fact (1), it lies in the center of and . Further, by the equality of operations:

Using that and that is in the center, this gives:

Multiplying both sides by (this is with respect to , the default multiplication):

We are now in a position to define the isomorphism . The isomorphism is:

The above shows that it is a homomorphism. It is clearly bijective, hence it is an isomorphism.