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Let $G$ be a group with identity denoted $1$. Let $\alpha$ denote an arbitrary element of $G$. The following two classes of group operations $f_\alpha : G \to G$ have the property that $f_\alpha(1) = 1$ for every $\alpha$.

  1. Conjugate the input by $\alpha$, i.e. $f_\alpha(x) = \alpha \cdot x \cdot \alpha^{-1}$.
  2. Form the commutator of the input and $\alpha$, i.e. $f_\alpha(x) = \alpha \cdot x \cdot \alpha^{-1} \cdot x^{-1}$.

Are there other common/useful/interesting (subjectively defined) classes of functions that are also identity-preserving?

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I don't quite see the point of this... –  Mariano Suárez-Alvarez Jan 28 '13 at 16:55
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Every group homomorphism preserves the identity. –  user58512 Jan 28 '13 at 17:00
    
This is a weird question, as user58512's comment show. Not to mention that the "subjectively defined" part is one of the weirdest ones, objectively speaking. –  DonAntonio Jan 28 '13 at 17:12
    
Must functions between groups, that we deal with, do preserve the identity. Why do you ask? –  Berci Jan 28 '13 at 17:38
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1 Answer

Note that the operation of a group $G$ on some object $X$ is just a group homomorphism from $G$ to the automorphism group of $X$. If $X$ is $G$ itself, two possibilities stand out:

  • View $X=G$ only as a set. Then there is no reason to restrict oneself to actions that leave $1$ fixed. In fact, $G$ operates on itself by left multiplication and this does not leave $1$ fixed (unless $G=1$)
  • View $X=G$ as a group. Then the automorphisms are group homomorphisms and as such leave $1$ fixed.

By the way, note that your second example is not even a group action. For example if $G$ is abelian and nontrivial, then $f_\alpha$ is not an automorphism of $G$.

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