A normal subgroup (also called an invariant subgroup) is a subgroup H of a group G such that all conjugates of every element in H belong to H. Algebraically, this condition is expressed as gHg−1=H for any element g in G.
Conditions: H is a subgroup of G; The condition must hold for all elements in H; The condition must hold for all elements g in G
A normal subgroup (also called an invariant subgroup) is a subgroup H of a group G such that all conjugates of every element in H belong to H. Algebraically, this condition is expressed as gHg−1=H for any element g in G.
Conditions: H is a subgroup of G; The condition must hold for all elements in H; The condition must hold for all elements g in G
We can traverse all elements of a group G by constructing cosets using a subgroup H. By multiplying every element g in G with the subgroup H (forming left cosets gH or right cosets Hg), the union of all distinct cosets covers every element of G exactly once.
We can traverse all elements of a group G by constructing cosets using a subgroup H. By multiplying every element g in G with the subgroup H (forming left cosets gH or right cosets Hg), the union of all distinct cosets covers every element of G exactly once.
We can intuitively understand a normal subgroup by visualizing the group G as an outer rectangle and the subgroup H as an inner rectangle. The condition gHg−1=H means that if you 'transport' the entire region H using an element g and then bring it back using g−1, the resulting region still lies completely within H.
Conditions: H is a subgroup of G; g is an element of G outside H
We can intuitively understand a normal subgroup by visualizing the group G as an outer rectangle and the subgroup H as an inner rectangle. The condition gHg−1=H means that if you 'transport' the entire region H using an element g and then bring it back using g−1, the resulting region still lies completely within H.
Conditions: H is a subgroup of G; g is an element of G outside H