In the study of purely infinite \(C^{*}\) -algebras, understanding the structure of normal subgroups plays a key role in revealing the deeper algebraic and geometric properties of these objects. Building on a previous work on the unitary groups of Cuntz algebras, we extend these results to symmetries, idempotents, and general linear groups. These findings offer new insights into the interplay between algebraic structures and operator theory, with implications for the broader study of infinite-dimensional algebras. Al-Rawashdeh proved that the unitary group of Cuntz algebra is normally generated by some non-trivial involution, or a normal subgroup of the unitary group of certain purely infinite \(C^{*}\) -algebras, contains all the \(*\) -symmetries (involutions) under some conditions. In this paper, we extend the results of involutions, projections and unitary groups to the case of symmetries, idempotents and general linear groups. We prove that a normal subgroup of the general linear group of some \(C^{*}\) -algebras, contains all the symmetries, under certain conditions.