[lpetrich]

First, a note on the classification of finite "simple" groups (Classification Theorem of Finite Groups -- from Wolfram MathWorld).

• Infinite families: 15
• Sporadic groups: 26

The proof of this theorem is some 15,000 pages spread throughout the mathematical literature.

The size of the largest sporadic group, the Monster group, is
246 * 320 * 59 * 76 * 112 * 133 * 17 * 19 * 23 * 29 * 31 * 41 * 47 * 59 * 71
= 808,017,424,794,512,875,886,459,904,961,710,757,00 5,754,368,000,000,000
≈ 8×1053.

-

Turning to Lie algebras, the "simple" ones are all known, and the "semisimple" ones are products of the simple ones.

• Infinite families: 4
• Exceptional algebras: 5

The simple ones have structures that can be graphed with "Dynkin diagrams". The proof of this classification is much shorter, but it is still a bit long for here. The families are, with their numbers of generators:

• A(n), SU(n+1): n*(n+2)
• B(n), SO(2n+1): n*(2n+1)
• C(n), Sp(2n): n*(2n+1)
• D(n), SO(2n): n*(2n-1)

The exceptional groups, with their numbers of generators:

• G2: 14
• F4: 52
• E6: 78
• E7: 133
• E8: 248

The smallest of these algebras have some isomorphisms:

• SO(2) ~ U(1)
• SO(3) ~ SU(2) ~ Sp(2)
• SO(4) ~ SU(2) * SU(2)
• SO(5) ~ Sp(4)
• SO(6) ~ SU(4)