Ambivalence is quotient-closed

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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)
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If G is an ambivalent group, and H is a normal subgroup of G, the quotient group G/H is also an ambivalent group.

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Given: An ambivalent group G, a normal subgroup H of G with quotient map \varphi:G \to G/H, and an element u \in G/H.

To prove: There exists an element v \in G/H such that vuv^{-1} = u^{-1}.


Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 Let g \in G be such that \varphi(g) = u. The quotient map \varphi:G \to G/H is, by definition, surjective. direct.
2 There exists h \in G such that hgh^{-1} = g^{-1}. G is ambivalent. direct from definition.
3 The element v = \varphi(h) works, i.e., vuv^{-1} = u^{-1}. We start with hgh^{-1} = g^{-1}, apply \varphi to both sides, and use the properties of homomorphisms.