If $f$ preserves identity and inverses, is it necessarily a group homomorphism?

Question

It is a well-known fact that for any group homomorphism $f\colon (G,*)\to (H,\circ)$ we have \begin{gather*} f(1_G)=1_H\\ f(x^{-1})=f(x)^{-1} \end{gather*}

What about the converse?

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Question. If $f$ is a function which preserves the identity and the inverses, is it necessarily a group homomorphism? Is it true if we additionally assume that $f$ is bijection? Is it true if $(G,*)=(H,\circ)$ (i.e., for functions - or bijective functions - $(G,*) \to (G,*))$?

Or maybe the question is better formulated like this: What are some nice/natural counterexamples to the above claims?

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There should be some easy-to-find counterexamples. For example, if we take a group $G$ such that each element is self-inverse, i.e., $x*x=e$ for each $x\in G$, then the condition about inverses is trivially true; it just says $f(x)=f(x)$. So we are left with the maps preserving the identity. Such maps (even bijections) are unlikely to be always homomorphisms.

Answer 1

Different group structures on the same set could have the same identity (nullary) and inversion (unary) operations. For example, one could define $g \circ h$ to be $h*g$ to get the so-called opposite group, and $\circ$ is not the same as $*$ unless $(G,*)$ is abelian. Nonetheless, the identity in $(G,\circ)$ is the same as the one in $(G,*)$, and so is inversion. This means that the identity map $(G,*) \to (G,\circ)$ preserves the identity and inversion but is not a homomorphism.

Answer 2

Suppose $G$ is a nonabelian Lie group with lie algebra $\mathfrak{g}$. Its exponential map $\exp:\mathfrak{g}\to G$ (viewing $\mathfrak{g}$ as a Lie group under addition) is not a homomorphism, but it intertwines with inversion. Indeed, not only does it intertwine with inversion, it intertwines with all powers: $\exp(nX)=\exp(X)^n$. Even beyond this, its restriction to any 1D subspace is a homomorphism!

If you're unfamiliar with Lie theory, take $\exp:(M_2(\mathbb{R}),+)\to\mathrm{GL}_2\mathbb{R}$, where $M_2(\mathbb{R})$ is the group of $2\times2$ real matrices under addition, $\mathrm{GL}_2\mathbb{R}$ is the group of invertible $2\times2$ real matrices under matrix multiplication, and the exponential function is defined by the usual power series, only applied to matrices instead of scalars (it always converges).