# CHAPTER 10

# SYMMETRY AND GROUPS

# 10.1 More About Groups

Recall that a set together with a binary operation · is called a *group* if the following conditions are satisfied:

**1.** is closed under the binary operation; that is, if

*x* and

*y* are elements of

, then so is

*x* ·

*y*.

**2.** The associative law holds. If

*x, y*, and

*z* are elements of

, then

**3.** There is an identity element

*e* in

. For every

*x* in

*, e* ·

*x* =

*x* ·

*e* =

*x*.

**4.** is closed with respect to inversion. For every member

*x* in

, there is another member

*x′* also in such that

*x* ·

*x′* =

*x′* ·

*x* =

*e*.

When dealing with groups in general, the binary operation ...