sentences:  In abstract algebra, the even integers form various algebraic structures that require the inclusion of zero.The fact that the additive identity (zero) is even, together with the evenness of sums and additive inverses of even numbers and the associativity of addition, means that the even integers form a group.Moreover, the group of even integers under addition is a subgroup of the group of all integers; this is an elementary example of the subgroup concept.