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Suppose I had groups $G_1 = \{e_1\}$ and $G_2 = \{e_2\}$; where both groups only have a single element (the identity element). What would the free product of the two groups $G = G_1 * G_2$ be?

At first; I reasoned that since the free product can be thought of as 'words constructible from elements of $G_1$ and $G_2$', then the group would simply be:

$$G = \{\ e_1\ ,\ e_2\ ,\ e_1e_2\ ,\ e_2e_1\ ,\ e_1e_2e_1\ \ , ... \}$$

But then I wondered what the identity element of $G$ is? Because $e_1e_1 = e_1$ and $e_2e_2 = e_2$, then both $e_1$ and $e_2$ are idempotent elements of $G$, but groups only have a single idempotent element (the identity). So this would mean that $e_1 = e_2 = e$, which doesn't really make sense. Am I thinking about free products completely wrong?

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    I suspect in the definition of free product you were given, the identity in one group collapses in the words. See: https://en.wikipedia.org/wiki/Free_product#Construction – 2017-02-15
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    ah; I've messed up. You're right – 2017-02-15
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    The answer is that the free product has a single element. In general, if $|G_1|=1$ then, for any group $G_2$, $G_1*G_2$ is isomorphic to $G_2$. – 2017-02-15

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