AMU 2026 — Mathematics PYQ

1. Identify the Group and Identity Element

• The given group is the additive group of integers, denoted as $(\mathbb{Z},+)$.

• In any group under addition ($+$), the identity element is $0$, because for any integer $a$:

  $$a+0=0+a=a$$

2. Definition of the Order of an Element

In an additive group, the order of an element $a$, denoted as $o(a)$, is defined as the smallest positive integer $n$ such that adding $a$ to itself $n$ times yields the identity element ($0$):

$$n\cdot a=\underset{n\text{ times}}{\underbrace{a+a+\cdots +a}}=0$$

If no such positive integer $n$ exists, the element $a$ is said to have infinite order (or order infinity).

3. Analyzing for a Non-Zero Element ($a\ne 0$)

Let's choose any non-zero integer $a\in \mathbb{Z}$ (where $a\ne 0$) and look for a positive integer $n\ge 1$ such that:

$$n\cdot a=0$$

Since both $n$ and $a$ are non-zero numbers:

• If a > 0, then multiplying it by a positive integer $n$ results in a strictly positive number (n \cdot a > 0). Thus, it can never equal $0$.

• If a < 0, then multiplying it by a positive integer $n$ results in a strictly negative number (n \cdot a < 0). Thus, it can never equal $0$.

Because there is absolutely no positive integer $n$ that can satisfy $n\cdot a=0$ for any non-zero integer, the order of $a$ must be infinite.

Conclusion

• The only element in $(\mathbb{Z},+)$ with a finite order is the identity element $0$ itself, which has an order of $1$ ($1\cdot 0=0$).

• Every other non-zero element $a\ne 0$ has an infinite order.

This directly matches option (a).