Let G be the group of real matrices \[ ( x , y ) = ( e x a m p ; 0 y a m p ; 1 ) ( x , y ∈ R ) . (x,y) = \left ( {\begin {array}{*{20}{c}} {{e^x}} & 0 \\ y & 1 \\ \end {array} } \right )\quad (x,y \in R). \] Every proper closed two-sided ideal of L 1 ( G ) {L^1}(G) is contained in a maximal modular two-sided ideal. The strong radical of L 1 ( G ) {L^1}(G) is the set of all f ∈ L 1 ( G ) f \in {L^1}(G) with ∫ f ( x , y ) d y = 0 \smallint f(x,y)\;dy = 0 for almost all x ∈ R x \in R . The strong structure spaces of L 1 ( G ) {L^1}(G) and L 1 ( R ) {L^1}(R) are homeomorphic.