In the context of group theory, gH represents the left coset of the subgroup H with respect to the element g, while Hg represents the right coset of H with respect to g. For a normal subgroup, these two sets are identical.
Conditions: H is a subgroup of G; g is an element of the group G
In the context of group theory, gH represents the left coset of the subgroup H with respect to the element g, while Hg represents the right coset of H with respect to g. For a normal subgroup, these two sets are identical.
Conditions: H is a subgroup of G; g is an element of the group G
A normal subgroup satisfies gH=Hg because its defining property is that it is invariant under conjugation, meaning gHg−1=H. By right-multiplying both sides of this equation by g, the g−1 and g cancel out, yielding the equality of the left coset gH and the right coset Hg.
Conditions: H is a normal subgroup of G; g is an element of the group G
A normal subgroup satisfies gH=Hg because its defining property is that it is invariant under conjugation, meaning gHg−1=H. By right-multiplying both sides of this equation by g, the g−1 and g cancel out, yielding the equality of the left coset gH and the right coset Hg.
Conditions: H is a normal subgroup of G; g is an element of the group G
No, replacing the representative element g with another element g' from the same coset does not yield a different coset. If g' belongs to the same coset as g (i.e., g′∈gH), then the coset generated by g' is identical to the coset generated by g (g′H=gH).
Conditions: g and g' are both elements of the group G; g' and g belong to the same coset
No, replacing the representative element g with another element g' from the same coset does not yield a different coset. If g' belongs to the same coset as g (i.e., g′∈gH), then the coset generated by g' is identical to the coset generated by g (g′H=gH).
Conditions: g and g' are both elements of the group G; g' and g belong to the same coset