Question

In three element group $\{e,a,b\}$ where $e$ is the identity, $a^5{b}^4$ is equal to

• A. $a$
• B. $e$
• C. $ab$
• D. $b$

Answer 1

The correct option is A $a$

Given group $G\equiv \{e,a,b\}$

It is given that $e$ is an identity.

Therefore only possibilities we have are

Either $a\cdot a=e$ and $b\cdot b=e$ .... (i)

Or $a\cdot b=b\cdot a=e$ ...(ii)

Case I:

$a\cdot a=e$ and $b\cdot b=e$

$a^5{b}^4=a\cdot a\cdot a\cdot a\cdot a\cdot b\cdot b\cdot b\cdot b$

$\Rightarrow a^5{b}^4=e\cdot a\cdot a\cdot a\cdot b\cdot b\cdot e$

$\Rightarrow a^5{b}^4=e\cdot a\cdot e$

$\Rightarrow a^5{b}^4=a$

Case II:

$a\cdot b=b\cdot a=e$

$a^5{b}^4=a\cdot a\cdot a\cdot a\cdot a\cdot b\cdot b\cdot b\cdot b$

$\Rightarrow a^5{b}^4=a\cdot a\cdot a\cdot a\cdot e\cdot b\cdot b\cdot b$

$\Rightarrow a^5{b}^4=a\cdot a\cdot a\cdot e\cdot b\cdot b$

$\Rightarrow a^5{b}^4=a\cdot a\cdot e\cdot b$

$\Rightarrow a^5{b}^4=a\cdot e$

$\Rightarrow a^5{b}^4=a$

Hence, both cases resulted in $a$.