Question

Q. Do the even integer form a group w.r.t addition.

Answer 1

Answer on Question #58450 – Math – Abstract Algebra

Question

Do the even integer form a group w.r.t. addition?

Solution

The set of even number (including zero and negative even numbers) is a subset of the integers. It is a group with respect to addition because it satisfies all group axioms:

1. The sum of even integers is also even integer (the closure under addition)

2. For all even integers $a, b$ and $c$, $(a + b) + c = a + (b + c)$. This property is known as associativity.

3. If $a$ is any even integer, then $0 + a = a + 0 = a$. Zero is called the identity element of addition because adding it to any integer returns the same integer.

4. For every even integer $a$, there is an integer $b$ such that $a + b = b + a = 0$. The integer $b$ is called the inverse element of the integer $a$ and is denoted by $-a$.

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