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subnormal subgroup (Definition)

Let $ G$ be a group, and $ H$ a subgroup of $ G$. Then $ H$ is a subnormal subgroup of $ G$ if there is a natural number $ n$ and subgroups $ H_0,\dots,H_n$ of $ G$ such that

$\displaystyle H=H_0\triangleleft H_1\triangleleft \cdots\triangleleft H_n=G,$
where $ H_i$ is a normal subgroup of $ H_{i+1}$ for $ i=0,\dots,n-1$.

Subnormality is a strictly weaker condition than normality, as normality of subgroups is not transitive.

We may write $ H\operatorname{sn}G$ or $ H\triangleleft \triangleleft \, G$ or $ H\!\trianglelefteq \trianglelefteq G$ to indicate that $ H$ is a subnormal subgroup of $ G$.

In a nilpotent group, all subgroups are subnormal.

Subnormal subgroups are ascendant and descendant.



"subnormal subgroup" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: subnormal series, characterization of finite nilpotent groups, normal subgroup, characteristic subgroup, fully invariant subgroup

Other names:  subinvariant subgroup, attainable subgroup
Also defines:  subnormal, subnormality
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Cross-references: descendant, ascendant, nilpotent group, normality of subgroups is not transitive, normal subgroup, natural number, subgroup, group
There are 6 references to this entry.

This is version 18 of subnormal subgroup, born on 2002-12-14, modified 2006-11-29.
Object id is 3756, canonical name is SubnormalSubgroup.
Accessed 4126 times total.

Classification:
AMS MSC: 20E15 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Chains and lattices of subgroups, subnormal subgroups)
 20D35 (Group theory and generalizations :: Abstract finite groups :: Subnormal subgroups)
Pending Errata and Addenda
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