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
Contents
Statement
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
.
Related facts
- Characterization of subgroup of neutral elements of reducible multiary group
- Groups giving same reducible multiary group are isomorphic
Proof
(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 | ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() |
By asociativity, we have ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | We can define an operation ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step (1) | Step-direct | |
3 | The operation defined in Step (2) is associative, i.e., ![]() ![]() |
![]() ![]() |
Step (2) | We write the operation as ![]() ![]() |
4 | The operation defined in Step (2) has identity element ![]() |
![]() ![]() |
Step (2) | ![]() ![]() ![]() ![]() |
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 ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | The operation defined in Step (2) gives a group structure on ![]() |
Steps (2), (3), (4), (5) | Step-combination direct | |
7 | For any elements ![]() ![]() ![]() |
![]() |
Steps (2), (6) | We prove this by induction on ![]() ![]() ![]() ![]() ![]() |
8 | We are done | Steps (2), (6), (7) | By Steps (2) and (6), we have a group structure on ![]() ![]() ![]() |
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.