Let's analyze the Cayley table for the set $S = \{a, b, c, d, e, f, g, h\}$ with the binary operation $*$.

Part B1: Identity Element

The identity element $e$ in a set $S$ with a binary operation $*$ is defined as the element that, when combined with any element $x \in S$ using the operation $*$, returns $x$ itself. In other words, if $e$ is the identity element, then: $e * x = x * e = x \quad \text{for all } x \in S$

Looking at the Cayley table:

• We see that for the row corresponding to $g$, the element $g$ returns the element itself when combined with any element: g * a &= a \\ g * b &= b \\ g * c &= c \\ g * d &= d \\ g * e &= e \\ g * f &= f \\ g * g &= g \\ g * h &= h \\ \end{align*}$$
• Similarly, for the column corresponding to $g$, the element $g$ is returned when combined with any element: a * g &= a \\ b * g &= b \\ c * g &= c \\ d * g &= d \\ e * g &= e \\ f * g &= f \\ g * g &= g \\ h * g &= h \\ \end{align*}$$

Thus, $g$ is the identity element for this set $S$ with operation $*$.

Part B2: Why $S$ is Not a Group

For a set with a binary operation to be a group, it must satisfy the following properties:

1. Closure: The operation must be closed on the set.
2. Associativity: The operation must be associative.
3. Identity Element: There must be an identity element (as found in B1).
4. Inverses: Every element must have an inverse within the set.

Let’s check if every element in $S$ has an inverse under the operation $*$.

For example, consider element $a$. Its row and column in the table give us: