Central factor satisfies image condition

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This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) satisfying a subgroup metaproperty (i.e., image condition)
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Statement

Statement with symbols

Suppose H is a central factor of a group G (in other words, HC_G(H) = G). Suppose \varphi:G \to K is a surjective homomorphism of groups. Then, \varphi(H) is a central factor of K.

Definitions used

Central factor

Further information: Central factor

A subgroup H of a group G is termed a central factor of G</mah> if it satisfies the following equivalent conditions:

* <math>HC_G(H) = G, where C_G(H) denotes the centralizer of H in G.

  • Every inner automorphism of G restricts to an inner automorphism of H.

Related facts

Similar facts about related properties

Proof

Proof in terms of centralizers

Given: A central factor H of a group G. A surjective homomorphism \varphi:G \to K.

To prove: \varphi(H)C_K(\varphi(H)) = K.

Proof: By the definition of homomorphism, if two elements commute in G, their images commute in K. Thus, the definition of centralizer yields:

\varphi(C_G(H)) \le C_K(\varphi(H)).

Taking the product of both sides with \varphi(H) yields:

\varphi(H)\varphi(C_G(H)) \subseteq \varphi(H)C_K(\varphi(H)).

By the definition of homomorphism, the left side is the same as \varphi(HC_G(H)), which is \varphi(G) (since H is a central factor of G). \varphi(G) = K by the assumption of surjectivity, so we get:

K \subseteq \varphi(H)C_K(\varphi(H)) \subseteq K.

This forces:

\varphi(H)C_K(\varphi(H)) = K

completing the proof.