5.1 Identity and Inverse Elements
Two further features of an operation matter as much as commutativity and associativity, and they are what turn a set with an operation into something you can solve equations in.
Definition 5.13 (Identity element). An element \(e\in G\) is an identity for the operation \(*\) if \[a*e=e*a=a \quad \text {for every } a\in G .\]
Definition 5.14 (Inverse element). Suppose \(G\) has an identity \(e\). An element \(b\in G\) is an inverse of \(a\in G\) if \[a*b=b*a=e .\]
Example 5.15. On \(\mathbb {Z}\) under addition the identity is \(0\), since \(a+0=0+a=a\) for every integer, and the inverse of \(a\) is \(-a\), since \(a+(-a)=0\). Every element has an inverse.
On \(\mathbb {Z}\) under multiplication the identity is \(1\). But the only elements with inverses are \(1\) and \(-1\): for any other integer \(a\), the number \(\frac {1}{a}\) is not an integer. Widen the set to \(\mathbb {Q}\setminus \{0\}\) and every element gains an inverse.
Note 5.16. That contrast is the whole reason the number systems were extended in chapter 2. “Can I undo this operation?” is the same question as ”does every element have an inverse?”, and each time the answer was no, a larger system was built in which it became yes. Subtraction is addition of an additive inverse; division is multiplication by a multiplicative inverse.
Note also that an identity is unique when it exists. If \(e\) and \(f\) were both identities then \(e=e*f=f\), using \(f\)’s property for the first equality and \(e\)’s for the second — so there is never a question of which identity is meant.
Example 5.17. Let \(G=\{1,-1,i,-i\}\) under ordinary multiplication of complex numbers. The operation can be set out in full as a Cayley table, with each entry the product of its row label and its column label:
| \(\times \) | \(1\) | \(-1\) | \(i\) | \(-i\) |
| \(1\) | \(1\) | \(-1\) | \(i\) | \(-i\) |
| \(-1\) | \(-1\) | \(1\) | \(-i\) | \(i\) |
| \(i\) | \(i\) | \(-i\) | \(-1\) | \(1\) |
| \(-i\) | \(-i\) | \(i\) | \(1\) | \(-1\) |
Everything can be read off the table. Every entry lies in \(G\), so the operation is closed. The table is symmetric about its main diagonal, so the operation is commutative. The row for \(1\) repeats the column headings, so \(1\) is the identity. And \(e=1\) appears exactly once in every row, so every element has an inverse: \(1^{-1}=1\), \((-1)^{-1}=-1\), \(i^{-1}=-i\) and \((-i)^{-1}=i\).
Note 5.18. A Cayley table is the most efficient way to answer these questions for a small set, and each property has a visible signature in it:
- closure — no symbol appears that is not one of the row labels;
- commutative — the table is symmetric about the main diagonal;
- identity — some row reproduces the column headings exactly, and the matching column reproduces the row headings;
- inverses — the identity appears in every row (and every column).
Associativity is the exception: it cannot be read off at a glance and has to be checked, which is why it is usually inherited from a known operation rather than verified by hand.
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