Difference between Kleene Closure in Group and in Alphabet

Question

Having a group $A = \{a, b\}$ and an alphabet $B = \{a, b\}$

Is there any difference between $A^*$ and $B^*$?

I think there is no difference, but I would like to be sure. Thanks in advance.

Answer 1

You are thinking of the notion of a free monoid. Given any set $S$, the free monoid generated by $S$ is the monoid $(S^*,\cdot,\varepsilon)$ such that $S^*$ is the set of all finite sequences of elements from $S$, where $\cdot$ is concatenation and $\varepsilon$ denotes the empty sequence.

Indeed $S^*$ is exactly the Kleene closure of $S$, if $S$ is viewed as an alphabet.

Answer 2

The difference is that in a group the set of generators $A = \{a, b\}$ may not be free. That is, two distinct words in $B^*$ may map to the same element in $\,A^*.\,$ For example, it could be that $\,a^2=b^2\,$ in the group. The point is that this can not happen in free groups and monoids.