Let (F1,A) and (F2,A) be two soft groups over G and (F1,A) be a soft subgroup of (F2,A).
If (N,A) be a soft subgroup of (F,A), then (N,A,τ) is a soft topological soft group over X and (N,A,τ) is a soft topological soft subgroup of (F,A,τ).
(F 1,A) is a soft subgroup (soft normal subgroup) of (F 2,A), and.
If f : G → K be a homomorphism, then (f(F1),A) and (f(F2),A) are both soft subgroups over K, and (f(F1),A) is a soft subgroup of (f(F2),A [6].
For each α ∈ A, (ϕ F 1) = ϕ α (F 1 is a subgroup of F 2. Hence, (ϕ F 1,A) is a soft subgroup of (F 2,A).
Also, (∩i∈IH i ) is a subgroup (normal subgroup) of F, for all α ∈ A. Hence, (∩i∈IH i,A) is a soft subgroup (soft normal subgroup) of (F,A).
Let (F,A) be a soft group over G and {(H i A); i ∈ I} is a nonempty family of soft subgroups (soft normal subgroups) of (F A), where I is an index set, and then ⋂ ~ i ∈ I ( H i, A ) Open image in new window is a soft subgroup (normal soft subgroup) of (F,A [6].
(ii) For each α ∈ A, ( ϕ − 1 F 2 ) = ϕ α − 1 ( F 2 Open image in new window is a subgroup of F 1. Hence, (ϕ −1 F 2,A) is a soft subgroup of (F 1,A), and then by Theorem 22, (ϕ −1 F 2,A,τ) is a soft topological soft group over X and ( ϕ − 1 F 2, A, τ ) ≤ ~ ( F 1, A, τ ) Open image in new window. .
(iii) If (F4,A) be a soft normal subgroup of (F2,A), then (α−1F4,A) is a soft normal subgroup of (α−1F2,A).
(ii) If (F3,A) be a soft normal subgroup of (F1,A), then (α F3,A) is a soft normal subgroup of (α F1,A).
Let (N,A) be a soft normal subgroup of (F,A), and then for each x ∈ A, the canonical mapping ϕ x : F x) → F x)/N x), given by ϕ x = ξ N x), ξ ∈ F x), is an onto homomorphism[7].
