Identity and inversesAQA A-Level Further Maths: Revision notes

A binary operation $*$ on a set $S$ combines any two elements of $S$ to give an element of $S$. An identity element $e\in S$ satisfies $a*e=e*a=a\quad\text{for every }a\in S.$ Under ordinary addition on $\mathbb{R}$ the identity is $0$; under multiplication it is $1$. The identity depends on both the set and the operation, so it must be found afresh each time.

To find $e$, solve $a*e=a$ for $e$. The answer must not depend on $a$, and it must lie in $S$. Then check $e*a=a$ as well, unless the operation is commutative. Example: $a*b=a+b-3$ on $\mathbb{R}$. $a+e-3=a$ gives $e=3$. Since $3+a-3=a$, the identity is $3$. Example: $a*b=a+b+ab$ on $\mathbb{R}\setminus\{-1\}$. $a+e+ae=a$ gives $e(1+a)=0$, so $e=0$ because $a\neq-1$. Then $0*a=a$ too, so $0$ is the identity and $0\in S$. Example with no identity: $a*b=a+2b$. $a*e=a$ gives $e=0$, but $0*a=2a\neq a$, so no identity exists.

When an identity $e$ exists, the inverse of $a$ is the element $a^{-1}\in S$ with $a*a^{-1}=a^{-1}*a=e.$ Solve $a*x=e$ for $x$ in terms of $a$, and check that $x\in S$. The inverse depends on $a$, unlike the identity. Example: $a*b=a+b-3$ with $e=3$: $a+x-3=3$ gives $a^{-1}=6-a$. So $7^{-1}=-1$, and $3$ is its own inverse. Example: $a\circ b=\frac{ab}{4}$ on $\mathbb{R}$ has $e=4$. $\frac{ax}{4}=4$ gives $a^{-1}=\frac{16}{a}$, so $8^{-1}=2$. The element $0$ has no inverse, because $0\circ x=0\neq4$.

On $\{1,2,3,4,5,6\}$ under multiplication modulo $7$ the identity is $1$. To find $a^{-1}$, look for $x$ with $ax\equiv1\pmod 7$: $2\times4=8\equiv1$, $3\times5=15\equiv1$, $6\times6=36\equiv1$. So $2\leftrightarrow4$, $3\leftrightarrow5$, and $1$ and $6$ are their own inverses. On $\{1,3,5,7\}$ under multiplication modulo $8$ the identity is $1$ and every element is its own inverse ($3\times3=9\equiv1$, $5\times5=25\equiv1$, $7\times7=49\equiv1$). Inverses solve equations: to solve $3x\equiv4\pmod7$, multiply by $3^{-1}=5$ to get $x\equiv20\equiv6$.

In a Cayley table (operation table) the identity is the element whose row and column both repeat the headings unchanged. The inverse of $a$ is found where $a$'s row meets the column holding the identity: that column's label is $a^{-1}$. If the identity does not appear in an element's row, that element has no inverse. Elements that are their own inverse have the identity on the leading diagonal of the table.