Normal subgroup

In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a <a href="/facts/Subgroup/r91mBgpa">subgroup</a> that is <a href="/facts/Invariant_(mathematics)/9TeNN2ed">invariant</a> under <a href="/facts/Inner_automorphism/awJPzhDa">conjugation</a> by members of the <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> of which it is a part. In other words, a subgroup 
 
 
 
 N
 
 
 {\displaystyle N}
 
 of the group 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is normal in 
 
 
 
 G
 
 
 {\displaystyle G}
 
 if and only if 
 
 
 
 g
 n
 
 g
 
 −
 1
 
 
 ∈
 N
 
 
 {\displaystyle gng^{-1}\in N}
 
 for all 
 
 
 
 g
 ∈
 G
 
 
 {\displaystyle g\in G}
 
 and 
 
 
 
 n
 ∈
 N
 .
 
 
 {\displaystyle n\in N.}
 
 The usual notation for this relation is 
 
 
 
 N
 ◃
 G
 .
 
 
 {\displaystyle N\triangleleft G.}

Normal subgroups are important because they (and only they) can be used to construct <a href="/facts/Quotient_group/HlwpHtjb">quotient groups</a> of the given group. Furthermore, the normal subgroups of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 are precisely the <a href="/facts/Kernel_of_a_homomorphism/GGSpUPMJ">kernels</a> of <a href="/facts/Group_homomorphism/sDguaNYs">group homomorphisms</a> with <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> 
 
 
 
 G
 ,
 
 
 {\displaystyle G,}
 
 which means that they can be used to internally classify those homomorphisms.
<a href="/facts/%25C3%2589variste_Galois/T0BPM1zP">Évariste Galois</a> was the first to realize the importance of the existence of normal subgroups.

Normal subgroup open-in-new

Normal subgroup