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## 'What's a Group?' printed from http://nrich.maths.org/

This problem is about infinite groups. We first meet groups when we
do simple number work and they are useful in much more general
applications. To use groups in higher mathematics we need a precise
definition.

A group is a set of elements together with a binary operation and
four defining properies (see the notes). Some groups are
commutative (Abelian) and other groups are not.

(a) Convince yourself that the set of integers with the operation
of addition satisfies the four given conditions and so forms a
group. Why does the set of natural numbers (positive integers) with
the operation of subtraction not form a group?

(b) Convince yourself that the set of positive rational numbers
with the operation of multiplication forms a group. Why does the
set of positive rational numbers with the operation of division not
form a group?

(c) Why does the set of positive integers with the operation of
multiplication not form a group?

(d) Why does the set of positive even integers with the operation
of multiplication not form a group?

(e) The set of integers with the operation * defined such that $m*n
= m + n + 1$ is a group. Find the identity element and the inverse
of the element $m$.

(f) The set of integers with the operation * defined by $m*n = m +
(-1)^m n $ is a group. Find the identity element and the inverse of
the element $m$.

(g) The set of all real numbers excluding only the number -1
together with the operation $x*y = xy + x + y$ is a group. Find the
identity and the inverse of the element $x$.

To read more about groups and properties of groups, which may help
you with this problem, see

Small
Groups. The article is about finite groups.