Introduction to sporadic groups for physicists

We describe the collection of finite simple groups, with a view to physical applications. We recall first the prime cyclic groups Zp and the alternating groups Altn > 4. After a quick revision of finite fields , q = pf, with p prime, we consider the 16 families of finite simple groups of Lie type. There are also 26 extra ‘sporadic’ groups, which gather in three interconnected ‘generations’ (with 5+7+8 groups) plus the pariah groups (6). We point out a couple of physical applications, including constructing the biggest sporadic group, the ‘Monster’ group, with close to 1054 elements from arguments of physics, and also the relation of some Mathieu groups with compactification in string and M-theory.