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If there is a group $\text{G}(z, +)$ where $z$ refers to the set of integers, and “$+$” is addition. Then which of the following are possible sub groups.

  1. (set of even numbers $, +)$
  2. (multiples of $3, +)$
  3. (set of odd numbers $, +)$
  4. (multiples of $5, +)$
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  1. $\text{P}$ is subset of $\text{G}$
     It is closed under $\text{G}$ identity and inverse exists.
    $\therefore \text{P}$ is a group as well or subset of $\text{G}$
    $\therefore \text{P}$  is  sub group of $\text{G}$.

    B. It is also group because identity $e = 0$
    $8$ inverse exist and subset of $\text{G}$
    $\therefore$ It is a sub group.

    C. (odd no, $+)$ it is not a group because odd $+$ odd $=$ even
    $\therefore$ closure property does not exist.

    D. Identity $= 0$ and inverse exist also closed
    It is a subset of $\text{G}$, thus sub group.
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