On Rough Subgroup of a Group

The articles [2], [3], [1], [4], and [5] provide the terminology and notation for this paper. For simplicity, we adopt the following rules: G denotes a group, A, B denote non empty subsets of G, N , H, H1, H2 denote subgroups of G, and x, a, b denote elements of G. Next we state a number of propositions: (1) For every normal subgroup N of G and for all elements x1, x2 of G holds x1 ·N · (x2 ·N) = (x1 · x2) ·N. (2) For every group G and for every subgroup N of G and for all elements x, y of G such that y ∈ x ·N holds x ·N = y ·N. (3) Let N be a subgroup of G, H be a subgroup of G, and x be an element of G. If x · N meets H, then there exists an element y of G such that y ∈ x ·N and y ∈ H. (4) For all elements x, y of G and for every normal subgroup N of G such that y ∈ N holds x · y · x−1 ∈ N. (5) For every subgroup N of G such that for all elements x, y of G such that y ∈ N holds x · y · x−1 ∈ N holds N is normal. (6) x ∈ H1 ·H2 iff there exist a, b such that x = a · b and a ∈ H1 and b ∈ H2. (7) Let G be a group and N1, N2 be strict normal subgroups of G. Then there exists a strict subgroupM of G such that the carrier ofM = N1 ·N2.