The E-normal structure of odd dimensional unitary groups

Abstract

Abstract. In this paper we define odd dimensional unitary groups U2n+1(R, ∆). These groups contain as special cases the odd dimensional general linear groups GL2n+1(R) where R is any ring, the odd dimensional orthogonal and symplectic groups O2n+1(R) and Sp 2n+1 (R) where R is any commutative ring and further the first author's even dimensional unitary groups U2n(R, Λ) where (R, Λ) is any form ring. We classify the E-normal subgroups of the groups U2n+1(R, ∆) (i.e. the subgroups which are normalized by the elementary subgroup EU2n+1(R, ∆)), under the condition that R is either a semilocal or quasifinite ring with involution and n ≥ 3. Further we investigate the action of U2n+1(R, ∆) by conjugation on the set of all E-normal subgroups.

Sandwich classification for O 2n+1(R) and U 2n+1(R,Δ) revisited

“…That pqprσ, ǫs˚0q . p1, 0qq˝a P Ω @a P Jp∆q follows from the previous theorem and [2,Lemma 63]. Suppose now that (4) holds for some odd form ideal pI, Ωq.…”

“…The groups U 2n`1 pR, ∆q include as special cases the even-dimensional unitary groups U 2n pR, Λq where n P N and pR, Λq is a form ring, the general linear groups GL n pRq where R is any ring and n ě 2 and further the symplectic groups Sp n pRq and the orthogonal groups O n pRq where R is a commutative ring and n ě 2. For details see [2,Example 15].…”

“…(a) In many interesting situations, N U 2n`1 ppR, ∆q, pI, Ωqq equals U 2n`1 pR, ∆q, for example the equality holds if Ω " Ω I min or Ω " Ω I max . But it can also happen that N U 2n`1 ppR, ∆q, pI, Ωqq ‰ U 2n`1 pR, ∆q, see [2,Example 38]. (b) EU 2n`1 pR, ∆q Ď N U 2n`1 ppR, ∆q, pI, Ωqq, see [2,Corollary 36].…”

“…But it can also happen that N U 2n`1 ppR, ∆q, pI, Ωqq ‰ U 2n`1 pR, ∆q, see [2,Example 38]. (b) EU 2n`1 pR, ∆q Ď N U 2n`1 ppR, ∆q, pI, Ωqq, see [2,Corollary 36].…”

The E-normal structure of Petrov’s odd unitary groups over commutative rings

“…Recall that a subgroup of a classical group G is called E-normal if it is normalized by the elementary subgroup of G. Further Bass showed that (2) holds true if R is an arbitrary ring and n is large enough with respect to the stable rank of R, including the case that R is semilocal and n ě 3. From that moment on the focus of research changed.…”