Direct factor over central subgroup
From Groupprops
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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
Definition
Definition with symbols
Suppose is a subgroup of a group
. We say that
is a direct factor over central subgroup of
if it satisfies the following equivalent conditions:
- There exists a subgroup
of
such that
is a central subgroup of
and
is a direct factor of the quotient group
.
- In the quotient map
, where
is the center of
,
is a direct factor of
.
Relation with other properties
Stronger properties
property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|
Direct factor | Join of direct factor and central subgroup|FULL LIST, MORE INFO | |||
Central subgroup | Join of direct factor and central subgroup|FULL LIST, MORE INFO | |||
Join of direct factor and central subgroup | |FULL LIST, MORE INFO |
Weaker properties
property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|
Normal subgroup |