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 and , as described above, are groups.

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# Definition

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

**Closure**

For all *a*, *b* in *G*, the result of the operation, , is also in G.

**Associativity**

For all *a*, *b* and *c* in *G*, .

**Identity element**

There exists an element *e* in *G* such that, for every element *a* in *G*, the equation 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 (or , if the operation is denoted “+”), such that , 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