Quick Calculation 15.1.2

A First Course in String Theory

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Recall that a group is a set which is closed under an associative multiplication; it contains an identity element, and each element has a multiplicative inverse. Verify that $\displaystyle{U(1)}$ and $\displaystyle{U(N)}$, as described above, are groups.

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Definition

A group is a set, G, together with an operation $\displaystyle{\bullet}$ (called the group law of G) that combines any two elements a and b to form another element, denoted $\displaystyle{a \bullet b}$ or $\displaystyle{ab}$. To qualify as a group, the set and operation, $\displaystyle{(G, \bullet)}$, must satisfy four requirements known as the group axioms:

Closure

For all a, b in G, the result of the operation, $\displaystyle{a \bullet b}$, is also in G.

Associativity

For all a, b and c in G, $\displaystyle{(a \bullet b) \bullet c = a \bullet (b \bullet c)}$.

Identity element

There exists an element e in G such that, for every element a in G, the equation $\displaystyle{e \bullet a = a \bullet e = a}$ holds. Such an element is unique, and thus one speaks of the identity element.

Inverse element

For each a in G, there exists an element b in G, commonly denoted $\displaystyle{a^{-1}}$ (or $\displaystyle{-a}$, if the operation is denoted “+”), such that $\displaystyle{a \bullet b = b \bullet a = e}$, where e is the identity element.

— Wikipedia on Group (mathematics)

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The axioms for a group are short and natural… Yet somehow hidden behind these axioms is the monster simple group, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists.

— Richard Borcherds in Mathematicians: An Outer View of the Inner World

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2019.07.28 Sunday ACHK