Assume we define an operator $$a\circ b = a+b+k, \\\forall a,b\in \mathbb Z$$
Can we prove that it together with range for $a,b$ is a group, for any given $k\in \mathbb Z$?
I have tried, and found that it fulfills all group axioms, but I might have made a mistake?
If it is a group, does it have a name?
My observations:
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Closure is obvious as addition of integers is closed.
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Identity If we take $e=-k$, then $a\circ e = a+k-k=a$
Verification $e\circ a = -k\circ a = -k+a+k=a$, as required.
- Inverse would be $a^{-1} = -a-2k$, which is unique.
Verification of inverse $aa^{-1} = a + (-a-2k)+k = -k = e$, as required.
- Associativity $(a\circ b) \circ c = (a + (b+k)) + (c + k)$.
We see everything involved is addition, which is associative, so we can remove parentheses and change order as we wish.
from Hot Weekly Questions - Mathematics Stack Exchange
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