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Let $\phi$ be an automorphism on a group $G$. If $\phi$ maps any one non-identity element in $G$ to itself, is $\phi$ necessarily the identity map? What if $G$ is cyclic?

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If "any" means "all", then yes. If "any" means "some", then in genreal no. –  Hagen von Eitzen Mar 7 '13 at 19:11
    
If $\phi(x)=x$ for all $x\neq e$, all it remains to check is $\phi(e)=e$. This is automatically satisfied by every homomorphism. You do not need $G$ to be cyclic for that. –  1015 Mar 7 '13 at 19:12
    
What about the cyclic group $Z/8Z$? There is the automorphism taking $x$ to $-x$, but 4 remains fixed. –  John Smith Mar 7 '13 at 19:16
    
@JohnSmith: The identity must stay fixed. –  azimut Mar 7 '13 at 19:27
    
@azimut: Zero is still going to zero. The group automorphism I give is clearly not the identity map but it does fix a non-zero element. So Zach's result is false for cyclic groups. –  John Smith Mar 7 '13 at 19:31

3 Answers 3

up vote 3 down vote accepted

What about the cyclic group $Z/8Z$? Negation is the automorphism taking x to −x. It leaves both 4 and 0 fixed. Therefore mapping a non-identity element to itself is not enough even in the case of cyclic groups.

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Let $H$ be any group and $G = H\times H$. Then $$\phi : G\to G, \quad (x,y) \mapsto (y,x)$$ is an automorphism of $G$. The elements $(x,x)$ are fixed under $\phi$. As long as $H$ is not the trivial group, $\phi\neq\operatorname{id}_G$ and there are always non-identity elements of the form $(x,x)$.

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If $G$ is cyclic then it is isomorphic to $\mathbb Z_n$ for $n\in\mathbb N$ or to $\mathbb Z$. The automorphism group on $\mathrm{Aut}(\mathbb Z_n)$ is isomorphic to $\mathbb Z_n^\times$ which is the multiplicative group of integers mod $n$ that are coprime to $n$. A $k\in\mathbb Z_n^\times$ corresponds to $\phi_k$ which maps the generator $1$ to $k$. If the map has a fixed point $l$, this implies $\exists m\in\mathbb N: kl=mn+l$, thus $n|(k-1)l$. Since $k-1<n$, $l$ has to satisfy $\mathrm{gcd}(n,l)>1$.

This means that an automorphism on $\mathbb Z_p$ for $p$ prime has no fixed point, whereas for arbitrary $n$ there is likely to be an automorphism with a fixed point.
For example, $n$ is always mapped to itself by every automorphism on $\mathbb Z_{2n}$.

On the other hand $\mathbb Z$ has only two automorphisms: identity and negation.

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