Ambivalence is quotient-closed
From Groupprops
This article gives the statement, and possibly proof, of a group property (i.e., ambivalent group) satisfying a group metaproperty (i.e., quotient-closed group property)
View all group metaproperty satisfactions | View all group metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for group properties
Get more facts about ambivalent group |Get facts that use property satisfaction of ambivalent group | Get facts that use property satisfaction of ambivalent group|Get more facts about quotient-closed group property
Statement
If is an ambivalent group, and
is a normal subgroup of
, the quotient group
is also an ambivalent group.
Related facts
Proof
Given: An ambivalent group , a normal subgroup
of
with quotient map
, and an element
.
To prove: There exists an element such that
.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | Let ![]() ![]() |
The quotient map ![]() |
direct. | ||
2 | There exists ![]() ![]() |
![]() |
direct from definition. | ||
3 | The element ![]() ![]() |
We start with ![]() ![]() |