Question

Which of the following is a subgroup of the group $G=\{2^n|n\in Z\}$ under multiplication?

• A. $\{5^n|n\in N\}$
• B. $\{3^n|n\in Z\}$
• C. $\{6^n|n\in N\}$
• D. $\{4^n|n\in Z\}$

Answer 1

Answer: (D)

$\{4^n|n\in Z\}$

The variable $Z$ means an integer value (negative as well positive integer)

and $n$ belongs to $Z$, this means that the power of $2$ can either be negative or positive integer.

$2^2=4$

$2^{-2}=\frac{1}{4}$

$2^4=16$

$2^{-4}=\frac{1}{16}$

Similarly, for a value $4^n$ can be written as $2^{2n}$

when $n=2$

so $4^2=16$

so $2^{2\times 2}=2^4=16$