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A cycle of a finite group G is a minimal set of elements {A^0,A^1,...,A^n} such that A^0=A^n=I, where I is the identity element. A diagram of a group showing every cycle in ...

A Lie group is called semisimple if its Lie algebra is semisimple. For example, the special linear group SL(n) and special orthogonal group SO(n) (over R or C) are ...

A simple group is a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group. Simple groups ...

An Abelian group is a group for which the elements commute (i.e., AB=BA for all elements A and B). Abelian groups therefore correspond to groups with symmetric multiplication ...

The Held group is the sporadic group He of order |He| = 4030387200 (1) = 2^(10)·3^3·5^2·7^3·17. (2) It is implemented in the Wolfram Language as HeldGroupHe[].

The Lyons group is the sporadic group Ly of order |Ly| = 51765179004000000 (1) = 2^8·3^7·5^6·7·11·31·37·67. (2) It is implemented in the Wolfram Language as LyonsGroupLy[].

The O'Nan group is the sporadic group O'N of order |O'N| = 460815505920 (1) = 2^9·3^4·5·7^3·11·19·31. (2) It is implemented in the Wolfram Language as ONanGroupON[].

The Rudvalis group is the sporadic group Ru of order |Ru| = 145926144000 (1) = 2^(14)·3^3·5^3·7·13·29. (2) It is implemented in the Wolfram Language as RudvalisGroupRu[].

The Suzuki group is the sporadic group Suz of order |Suz| = 448345497600 (1) = 2^(13)·3^7·5^2·7·11·13. (2) It is implemented in the Wolfram Language as SuzukiGroupSuz[].

The Thompson group is the sporadic group Th of order |Th| = 90745943887872000 (1) = 2^(15)·3^(10)·5^3·7^2·13·19·31. (2) It is implemented in the Wolfram Language as ...