[Math] Example of an infinite group where every element except identity has order 2

group-theoryinfinite-groups

Find an infinite group, in which every element g not equal identity (e) has order 2

Does this question mean this:

the group that fail condition (2) which is no inverse and also that group must have the size 2

My answer:

Z*

Best Answer

No, that’s clearly not what it means: a group of size $2$ is not an infinite group. You’re to find an infinite group $G$ in which every element except the identity has order $2$, meaning that if $g\in G$, and $g$ is not the identity element $1_G$ of $G$, then $g^2=1_G$. Of course $1_G^2=1_G$ as well, so your problem is really to find an infinite group $G$ in which every element satisfies the equation $x^2=1_G$, where $1_G$ is the identity element in $G$.

HINT: First find a finite group $H$ with this property, and then look at the product of infinitely many copies of $H$.

Alternative HINT: Consider the operation of symmetric difference on the set of subsets of some infinite set.

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