Group powered over a unital ring

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Definition

Suppose G is a group and R is a unital ring. A structure of G as a group powered over R includes an operation G \times R \to G denoted by exponentiation, i.e., the output of (g,r) is denoted g^r, satisfying the following conditions:

  • For r an integer (modulo whatever is the characteristic of the ring), g^r is the usual g^r.
  • For r,s \in R, (g^r)^s = g^{rs}.
  • For r,s \in R, (g^r)(g^s) = g^{r + s}

As a variety of algebras

For any fixed unital ring R, the groups powered over R form a variety of algebras. This variety admits the variety of groups as a reduct, i.e., every group powered over a unital ring gives rise to a group.

Functors

Suppose R and S are unital rings and \varphi:R \to S is a homomorphism of unital rings. Then, if G is a group equipped with a powering structure over S, we naturally get a powering structure of G over R.

Particular cases

Cases based on rings

Value of ring Notion of group powered over that ring Is the powering uniquely determined by the abstract group structure? Does every group admit a powering over such a ring? Note that this condition would hold if the ring has a surjective ring homomorphism to \mathbb{Z}.
\mathbb{Z} usual concept of group Yes Yes
\mathbb{Z}[\pi^{-1}] where \pi is a set of primes group powered over a set of primes where the set of primes is \pi. In other words, every element of the group has a unique p^{th} root in the group. Yes No (unless \pi is empty)
\mathbb{Q} rationally powered group: unique p^{th} roots for all primes p Yes No
\mathbb{Z}/n\mathbb{Z} for a positive integer n group whose exponent is finite and divides n Yes No
\mathbb{Z}[t] group along with a set map from the group to itself (need not be a homomorphism) that commutes with powering and sends every element to an element that it commutes with. No Yes
\mathbb{Q}[t] rationally powered group along with a set map from the group to itself that commutes with powering and sends every element to an element that commutes with it. No No

Unipotent algebraic groups

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