TITLE: Does the set of all strings of letters form a group?
QUESTION [6 upvotes]: This is related to a course I'm taking in computer science theory. 
Let $\sum$ be an alphabet.  Then the set of all strings over $\sum$, denoted as $\sum^*$ has the operation of concatenation (adjoining two strings end to end).  Clearly, concatenation is associative, $\sum^*$ is closed under concatenation, and the identity element is the empty string.  I'm also taking a course in modern algebra, so I naturally ask can $\sum^*$ be formed into a group?  Three of the four group axioms are satisfied.

REPLY [2 votes]: No, $\Sigma^*$ is not a group (unless $\Sigma = \emptyset $, in which case $\Sigma ^*$ is a trivial group with one element). The reason is that the only element having an inverse is the empty word. So if $a\in \Sigma$, then $a$ as an element of $\Sigma ^*$ does not have an inverse (rigorously, note that the length of words never decreases upon concatenation). 
What $\Sigma ^*$ is is an example of a monoid. A monoid is an algebraic structure satisfying all axioms of a group except the requirement of identities. In fact, $\Sigma ^*$ is the free monoid on the set $\Sigma$. There is also the obvious notion of a commutative monoid, and a certain quotient of $\Sigma^*$, obtained by allowing elements to commute, gives rise to the free commutative monoid $\Sigma ^+$.