Hello, I would really appreciate any help with the following problem:

Prove that

is a group, where

and

,

.... .

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To prove that this is a group,

must satisfy closure, associativity, existence of an identity element and existence of inverse elements.

Could the property of associativity be demonstrated like this?

Therefore,

, so the property of associativity is satisfied.

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Is the identity element

?

From

it follows

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Is the inverse element of

,

?

Because

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If the above is correct (and I'm not certain that it is), the only property left to be demonstrated is closure, i.e. if

then

.

And here I need your help! How to demonstrate the property of closure?

There is a hint in the text: "Observe that

, and prove that

if and only if

and

".

So, how to prove the statements from the hint, and, more importantly, how to apply them to demonstrate the closure property of

?

Many thanks! :)