Inverse map is involutive

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This article gives the statement, and possibly proof, of a basic fact in group theory.
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Statement

The inverse map in a group, i.e. the map sending any element of the group, to its inverse element, is an involutive map, in the sense that it has the following two properties:

  • It satisfies the reversal law:

(a_1a_2 \ldots a_n)^{-1} = a_n^{-1}a_{n-1}^{-1}\ldots a_1^{-1}

  • Applying it twice sends an element to itself:

(a^{-1})^{-1} = a

It fact, both these are true in the greater generality of a monoid, under the condition that all the a_is have two-sided inverse a_i^{-1} (note: we still need a monoid to guarantee that two-sided inverses, when they exist, are unique).

Proof

Proof of reversal law

In order to show that the element a_n^{-1}a_{n-1}^{-1}\ldots a_1^{-1} is a two-sided inverse of a_1a_2 \ldots a_n, it suffices to show that their product both ways is the identity element. Consider first the product:

(a_1a_2 \ldots a_{n-1}a_n)(a_n^{-1}a_{n-1}^{-1}\dots a_2^{-1}a_1^{-1})

Due to associativity, we can drop the parentheses and we get:

a_1a_2 \ldots a_{n-1}a_na_n^{-1}a_{n-1}^{-1}\dots a_2^{-1}a_1^{-1}

Now, consider the middle product a_na_n^{-1}. This is the identity element, and since the identity element has no effect on the remaining product, it can be removed, giving the product:

a_1a_2 \ldots a_{n-1}a_{n-1}^{-1}\dots a_2^{-1}a_1^{-1}

We now repeat the argument with the middle product a_{n-1}a_{n-1}^{-1} and cancel them. Proceeding this way, we are able to cancel all terms and eventually get the identity element.

A similar argument follows for the product the other way around:

(a_n^{-1}a_{n-1}^{-1}\ldots a_1^{-1})(a_1a_2 \ldots a_n)

Thus, the elements are two-sided inverses of each other.

Note: In fact, it suffices to check only one of the two inverse conditions, i.e., check only that the first product is the identity element. This is because, in a group, every element has a two-sided inverse. Further, equality of left and right inverses in monoid forces any one-sided (left or right) inverse to be equal to the two-sided inverse.

Proof for applying it twice

This is direct from the definition. let b = a^{-1}. Then, by the inherent symmetry in the definition of inverse element, we see that a = b^{-1}.

More explicitly, if b = a^{-1}, that means that ab = ba = e. But this is precisely the condition for stating that a = b^{-1}.


References

Textbook references

  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Page 18, Proposition 1