Definition·D048
Normal subgroup
A subgroup closed under conjugation by every element of the group.
A subgroup
of a group
is normal, written
, when it is closed under conjugation by every group element:
The element
is the conjugate of
by
.
In words
For any g and n, if g is in G and n is in N then the conjugate g n g⁻¹ is in N too.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Equivalently the left and right cosets agree,
for every
. This conjugation-invariance is exactly the condition making the coset quotient
into a group under
, the quotient group to be built next. Both trivial subgroups
and
are normal, and in an abelian group every subgroup is normal. The normal subgroups are precisely the kernels of homomorphisms (L39).