On Pro-$2$ Identities of $2\times2$ Linear Groups
David El-Chai Ben-Ezra, Efim Zelmanov

TL;DR
This paper proves that the pro-2 congruence subgroup of $GL_2( ext{complete local ring})$ admits a pro-2 identity, extending previous results known for odd primes using trace identities from PI-theory.
Contribution
It establishes the existence of a pro-2 identity for $GL_2^1( ext{ring})$ when the ring's characteristic is 2, filling a gap in the understanding of pro-$p$ identities.
Findings
Pro-2 group $GL_2^1( ext{ring})$ admits a pro-2 identity when $ ext{char}( ext{ring})=2$.
Extension of Zubkov's result from $p eq 2$ to $p=2$ case.
Use of trace identities from PI-theory to prove the existence of pro-$p$ identities.
Abstract
Let be a free pro- non-abelian group, and let be a commutative Noetherian complete local ring with a maximal ideal such that . In [Zu], Zubkov showed that when , the pro- congruence subgroup admits a pro- identity, i.e., there exists an element that vanishes under any continuous homomorphism . In this paper we investigate the case . The main result is that when , the pro- group admits a pro- identity. This result was obtained by the use of trace identities that originate in PI-theory.
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Taxonomy
TopicsFinite Group Theory Research · Advanced Topics in Algebra · Algebraic Geometry and Number Theory
