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## Homework Statement

Let G be a group. Assume a to be an element of the group. Then the set <a> = {a

^{k}I k∈ℤ} is a subgroup of G.

I am confused as to why the proof makes the assumption that <a> is a subset of the set G.

## Homework Equations

## The Attempt at a Solution

The proof I think is like the following:

As the identity element is in G it is true that it is also in <a>. Since for a general group G, the inverse is denoted as a

^{0}. Let b=a

^{r}and c=a

^{j}be elements in the group then a

^{r}a

^{j}is an element in the group due to the axiom of exponents... Now proving that there exists an inverse is done in an similar way but even though these three conditions are satisfied, shouldn't I prove that <a> is a subset of G?[/B]