profinite group

English

Alternative forms

• pro-finite group

Noun

profinite group (plural profinite groups)

1. (topology, category theory) A topological group that is isomorphic to the inverse limit of some inverse system of discrete finite groups; equivalently, a topological group that is also a Stone space.
   • 2000, Haruzo Hida, Modular Forms and Galois Cohomology, Cambridge University Press, page 28:

     This proposition shows that to have a representation with big image of a profinite group, for example, a Galois group, we need to take a profinite topological ring $A$ as a coefficient ring, like the p-adic integer ring $\mathbb {Z} _{p}$.

   • 2011, Moshe Jarden, Algebraic Patching, Springer, page 207:

     We have already pointed out that a profinite group G of an infinite rank m is free of rank m if (and only if) G is projective and every finite split embedding problem for G with a nontrivial kernel has m solutions (Proposition 9.4.7). Dropping the condition on G to be projective leads to the notion of a "quasi-free profinite group" (Section 10.6).

   • 2014, David Barnes, Constanze Roitzheim, Rational equivariant rigidity, Christian Ausoni, Kathryn Hess, Brenda Johnson, Wolfgang Lück, Jérôme Scherer, An Alpine Expedition through Algebraic Topology, American Mathematical Society, page 14,

     Recall that a profinite group is an inverse limit of an inverse system of finite groups, with the p-adic numbers being the canonical example. Any finite group is, of course, profinite, but when we talk of a profinite group we assume that the group is infinite.

Related terms

• profinite
• profinite integer
• profinite word

Translations

topological group isomorphic to the inverse limit of an inverse system of discrete finite groups

• French: groupe profini m
• German: proendliche Gruppe f

Further reading

• Profinite group on Encyclopedia of Mathematics
• profinite group on nLab