Zp×

Zp× is a notation that can refer to two related but distinct multiplicative groups, depending on the mathematical context. In elementary number theory, it usually denotes the multiplicative group of units modulo p, written as (Z/pZ)×. In p-adic number theory, it denotes the unit group of the p-adic integers, Zp×. Both are denoted with a similar symbol but have different structures and interpretations.

Finite field case (Z/pZ)×. When p is prime, Z/pZ is the finite field with p elements. The group of nonzero residues under multiplication modulo p, (Z/pZ)×, has order p−1 and is abelian and cyclic. Therefore there exists a primitive root g modulo p that generates all nonzero residues: (Z/pZ)× = ⟨g⟩. For example, modulo 7, the element 3 is a primitive root, generating all six nonzero residues. For p = 2, the group is trivial, consisting only of 1.

p-adic case (Zp×). In the p-adic setting, Zp denotes the ring of p-adic integers and Zp× its group of units. For odd p, Zp× has a canonical structure Zp× ≅ μ_{p−1} × (1 + pZp), where μ_{p−1} is the cyclic group of (p−1)th roots of unity and 1 + pZp is a pro-p group isomorphic to Zp. Thus Zp× is a compact, totally disconnected topological group. For p = 2, Z2× ≅ C2 × Z2, i.e., a cyclic group of order 2 times a copy of Z2.

Notes. The notation Zp× is overloaded; the meaning is determined by context—modulo p versus p-adic units. In either case, the group is central to topics in number theory, primitive roots, and p-adic analysis.