Not normal subgroup of

A normal subgroup stays the same under conjugation. A group is a mathematical set with an operation, such as composing symmetries, that follows specific rules. A subgroup is a group within it, using the same operation.

Conjugation replaces each element h of a subgroup H with g h g⁻¹, where g is any element of the containing group G and g⁻¹ is its inverse. H is normal when this always gives back the same subgroup as a whole; individual elements may move. If H is a subgroup but this condition fails, H ⋪ G.

Normality is what allows the cosets of H—the sets obtained by multiplying H by elements of G—to form the quotient group G/H with the operation inherited from G.