**A.A. Ambily**

TIFR

March 12, 2015

**Normality and $K_1$-stability of Roy's elementary orthogonal group**:
A. Roy introduced the elementary orthogonal group $EO_A(Qperp H(A)^m)$ of a quadratic space with a hyperbolic summand over a commutative ring $A$. This construction of Roy generalized the earlier work's of Dickson-Siegel-Eichler-Dieudonn{e} over fields.
In this talk, we shall discuss the normality of the elementary orthogonal group (Dickson--Siegel--Eichler--Roy or DSER group) $EO_A(Qperp H(A)^m)$ under some conditions on the hyperbolic rank. We also establish stability results for $K_1$ of Roy's elementary orthogonal group under different stable range conditions. The stability problem for $K_1$ of quadratic forms was studied in 1960's and in early 1970's by H. Bass, A. Bak, A. Roy, M. Kolster and L.N. Vaserstein. We obtain a Dennis-Vaserstein type decomposition theorem for the elementary orthogonal group (DSER group) which is used to deduce the stability theorem.