Subnormal subgroup

In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G.

In notation,

H

{\displaystyle H}

is

k

{\displaystyle k}

-subnormal in

G

{\displaystyle G}

if there are subgroups

H

=

H

0

,

H

1

,

H

2

,

…

,

H

k

=

G

{\displaystyle H=H_{0},H_{1},H_{2},\ldots ,H_{k}=G}

of

G

{\displaystyle G}

such that

H

i

{\displaystyle H_{i}}

is normal in

H

i

+

1

{\displaystyle H_{i+1}}

for each

i

{\displaystyle i}

.

A subnormal subgroup is a subgroup that is

k

{\displaystyle k}

-subnormal for some positive integer

k

{\displaystyle k}

.

Some facts about subnormal subgroups:

A 1-subnormal subgroup is a proper normal subgroup (and vice versa).

A finitely generated group is nilpotent if and only if each of its subgroups is subnormal.

Every quasinormal subgroup, and, more generally, every conjugate-permutable subgroup, of a finite group is subnormal.

Every pronormal subgroup that is also subnormal, is normal. In particular, a Sylow subgroup is subnormal if and only if it is normal.

Every 2-subnormal subgroup is a conjugate-permutable subgroup.

The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal

subgroup is subnormal. The relation of subnormality can be defined as the transitive closure of the relation of normality.