# Group: Exercises

# Preliminaries

Show that the identity element in a group is unique. That is, if \(G\) is a group and two elements \(e_1, e_2 \in G\) both satisfy the axioms describing the identity element, then \(e_1 = e_2\).

Show that inverses are also unique. That is, if \(g \in G\) is an element of a group and \(h_1, h_2 \in G\) both satisfy the axioms describing the inverse of \(g\), then \(h_1 = h_2\).

and, on the other hand,

Hence \(h_1 = h_2\). <div><div>

# Examples involving numbers

Determine whether the following sets equipped with the specified binary operations are groups. If so, describe their identity elements (which by the previous exercise must be unique) and how to take inverses.

The real numbers \(\mathbb{R}\) together with the addition operation \((x, y) \mapsto x + y\).

The real numbers \(\mathbb{R}\) together with the multiplication operation \((x, y) \mapsto xy\).

The positive real numbers \(\mathbb{R}_{>0}\) together with the multiplication operation \((x, y) \mapsto xy\).

The real numbers \(\mathbb{R}\) together with the operation \((x, y) \mapsto x + y - 1\).

The real numbers \(\mathbb{R}\) together with the operation \((x, y) \mapsto \frac{x + y}{1 + xy}\).

This operation is interesting and useful, though, when it is defined. It shows up in special relativity, where it describes how velocities add relativistically (in units where the speed of light is \(1\)). <div><div>

Parents:

- Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.