AMITâS CONJECTURE FOR WORDS OVER TWO VARIABLES - Siddiq Wira Awaldy, Muhammad
Algebraic structure introduced us to some mathematical structures, and one of the structure is a group. Unlike the usual number system where the multiplication always commute, one can construct a group that does not necessarily commute. To measure the commutativity of two elements in a group, we can define commutator by [x; y] = x????1y????1xy for all x; y 2 G. Then one can say that 2 elements x and y in G are commute if and only if [x; y] = e where e is an identity element of G. One could ask ”what is the probability of two elements x and y in a finite group such that [x; y] = e?” Then what if we replace [x; y] with another expression? What if we replace e with another element in G? Using free group, we can define a probability associated with verbal subgroup of G denoted as PG;w(g). There are many open problems about structure of PG;w(g), and its implication to the underlying group structure. One of them is Amit’s Conjecture which said the value of PG;w(e) never be less than 1 jGj for every finite nilpotent group G. In this final project, we will proof Amit’s Conjecture for any words over two variable.