Group/Examples/x+y+2 over Reals

Example of Group

Let $\circ: \R \times \R$ be the operation defined on the real numbers $\R$ as:

$\forall x, y \in \R: x \circ y := x + y + 2$

Then $\struct {\R, \circ}$ is a group whose identity is $-2$.

Proof

Taking the group axioms in turn:

Group Axiom $\text G 0$: Closure

$\forall x, y \in \R: x + y + 2 \in \R$

Thus $x \circ y \in \R$ and so $\struct {\R, \circ}$ is closed.

$\Box$

Group Axiom $\text G 1$: Associativity

\(\ds x \circ \paren {y \circ z}\)

\(=\)

\(\ds x + \paren {y + z + 2} + 2\)

\(\ds \)

\(=\)

\(\ds x + y + z + 4\)

\(\ds \paren {x \circ y} \circ z\)

\(=\)

\(\ds \paren {x + y + 2} + z + 2\)

\(\ds \)

\(=\)

\(\ds x + y + z + 4\)

\(\ds \)

\(=\)

\(\ds x \circ \paren {y \circ z}\)

Thus $\circ$ is associative.

$\Box$

Group Axiom $\text G 2$: Existence of Identity Element

Let $y$ be such that $x \circ y = x$.

Then:

\(\ds x \circ y\)

\(=\)

\(\ds x\)

\(\ds \leadsto \ \ \)

\(\ds x + y + 2\)

\(=\)

\(\ds x\)

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds -2\)

Then it is noted that:

\(\ds -2 \circ x\)

\(=\)

\(\ds -2 + x + 2\)

\(\ds \)

\(=\)

\(\ds x\)

Thus $-2$ is the identity element of $\struct {\R, \circ}$.

$\Box$

Group Axiom $\text G 3$: Existence of Inverse Element

We have that $-2$ is the identity element of $\struct {\R, \circ}$.

So:

\(\ds x \circ y\)

\(=\)

\(\ds -2\)

\(\ds \leadsto \ \ \)

\(\ds x + y + 2\)

\(=\)

\(\ds -2\)

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds -x - 4\)

Then it is noted that:

\(\ds \paren {-x - 4} \circ x\)

\(=\)

\(\ds -x - 4 + x + 2\)

\(\ds \)

\(=\)

\(\ds -2\)

Thus every element of $\struct {\R, \circ}$ has an inverse $-x - 4$.

$\Box$

All the group axioms are thus seen to be fulfilled, and so $\struct {\R, \circ}$ is a group.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Exercise $1 \ \text{(b)}$