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have been proved in 1994 by A. Shalev and independently by C. R. Leedham-Green. Theorem D is one of these theorems and asserts that, for any prime number
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elements. More generally it was shown that a finitely generated profinite group is a compact p-adic
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402:{\displaystyle \mathbb {Z} _{p}=\displaystyle \varprojlim \mathbb {Z} /p^{n}\mathbb {Z} .}
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such that group multiplication and inversion are both analytic functions. The work of
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Fact: A finite homomorphic image of a pro-p group is a p-group. (due to J.P. Serre)
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if and only if it has an open subgroup that is a uniformly powerful pro-p-group.
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such that any closed subgroup has a topological generating set with no more than
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564:-adic analytic groups mentioned above can all be found as closed subgroups of
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305:. This finiteness result is fundamental for the classification of finite
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The best-understood (and historically most important) class of pro-
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analytic groups: groups with the structure of an analytic
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513:consisting of all matrices congruent to the
234:-adic analytic if and only if it has finite
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608:{\displaystyle \ GL_{n}(\mathbb {Z} _{p})}
459:{\displaystyle \ GL_{n}(\mathbb {Z} _{p})}
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147:Alternatively, one can define a pro-
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797:. You can help Knowledge (XXG) by
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502:{\displaystyle \ \mathbb {Z} _{p}}
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144:quotient group is always finite.
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226:-adic numbers, shows that a pro-
203:{\displaystyle \mathbb {Q} _{p}}
83:{\displaystyle N\triangleleft G}
647:Residual property (mathematics)
669:; Mann, A.; Segal, D. (1991),
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322:The canonical example is the
701:New Horizons in pro-p Groups
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654:(See Property or Fact 5)
293:and any positive integer
214:and Mann, combined with
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220:Hilbert's fifth problem
793:-related article is a
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560:group. In fact the
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114:{\displaystyle G/N}
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271:{\displaystyle r}
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327:-adic integers
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622:Any finite
18:mathematics
838:Categories
659:References
415:The group
28:(for some
556:is a pro-
368:
363:←
280:Lie group
230:group is
222:over the
163:-groups.
75:◃
854:P-groups
641:See also
475:matrices
317:Examples
212:Lubotzky
179:manifold
142:discrete
693:1152800
552:. This
517:modulo
130:compact
35:) is a
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627:-group
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155:of an
134:closed
126:-group
789:This
732:This
477:over
181:over
175:-adic
138:index
121:is a
26:group
795:stub
738:stub
705:ISBN
679:ISBN
285:The
236:rank
90:the
60:open
22:pro-
20:, a
470:by
360:lim
16:In
840::
689:MR
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313:.
826:e
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812:v
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625:p
619:,
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603:)
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588:(
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495:p
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472:n
468:n
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439:(
434:n
430:L
426:G
396:.
392:Z
386:n
382:p
377:/
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354:=
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299:p
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196:p
191:Q
173:p
168:p
161:p
149:p
124:p
109:N
105:/
101:G
78:G
72:N
46:G
33:p
24:p
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