(Theorem:19)(part-1) If H is a subgroup of a group G and a,b€G then Ha=Hb iff ab^-1€H

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Let H be subgroup of a group G and let a, b∈G.Then i) aH =bH iff a^-1b∈ H ii)Ha=Hb iff ab^-1∈H Let H be subgroup of a group G and let a, b∈G.Then i) aH =bH iff a^-1b∈ H ii)Ha=Hb iff ab^-1∈H
A subgroup H of a group G is a normal subgroup of G if and only if ghg^-1 ∈ H ∀ g∈G and h∈H. A subgroup H of a group G is a normal subgroup of G if and only if ghg^-1 ∈ H ∀ g∈G and h∈H.
2. Properties of Cosets in Group Theory | Ha = H iff a ∈ H | Ha = Hb iff ab^-1 ∈ H | Proof 2. Properties of Cosets in Group Theory | Ha = H iff a ∈ H | Ha = Hb iff ab^-1 ∈ H | Proof
properties of cosets | Ha =H iff a belongs to H | Ha = Hb iff ab(inverse) belongs to H #groups properties of cosets | Ha =H iff a belongs to H | Ha = Hb iff ab(inverse) belongs to H #groups
A non-empty subset H of a group G is a subgroup of G if and only if a*b^(-1)∈H for all a,b ∈H. A non-empty subset H of a group G is a subgroup of G if and only if a*b^(-1)∈H for all a,b ∈H.
If H is a subgroup of G and aH=bH implies Ha=Hb then gHg^-1 is a subset of H . If H is a subgroup of G and aH=bH implies Ha=Hb then gHg^-1 is a subset of H .
A non-empty subset H of a group G Is a subgroup of G if and only if ab inverse ∈ H | NERDY CREW A non-empty subset H of a group G Is a subgroup of G if and only if ab inverse ∈ H | NERDY CREW
(Theorem:17) part-1(If H is a subgroup of a group G then for any a,b€G (a)aH~bH ) (Theorem:17) part-1(If H is a subgroup of a group G then for any a,b€G (a)aH~bH )