Element structure of projective special linear group:PSL(3,2)
This article gives specific information, namely, element structure, about a particular group, namely: projective special linear group:PSL(3,2).
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This article describes the element structure of projective special linear group:PSL(3,2), which is the same as , . It is also isomorphic to the projective special linear group of degree two over field:F7, i.e., the group .
|order of the whole group (total number of elements)||168|
|conjugacy class sizes|| 1,21,24,24,42,56|
in grouped form: 1 (1 time), 21 (1 time), 24 (2 times), 42 (1 time), 56 (1 time)
maximum: 56, number of conjugacy classes: 6, lcm: 168
|order statistics|| 1 of order 1, 21 of order 2, 56 of order 3, 42 of order 4, 48 of order 7|
maximum: 7, lcm (exponent of the whole group): 84
Conjugacy class structure
Interpretation as projective special linear group of degree two
Compare with element structure of projective special linear group of degree two over a finite field#Conjugacy class structure
We consider the group as , . We use the letter to denote the generic case of .
|Nature of conjugacy class upstairs in||Eigenvalues||Characteristic polynomial||Minimal polynomial||Size of conjugacy class (generic that is 3 mod 4)||Size of conjugacy class ()||Number of such conjugacy classes (generic that is 3 mod 4)||Number of such conjugacy classes ()||Total number of elements (generic that is 3 mod 4)||Total number of elements ()|
|Diagonalizable over with equal diagonal entries, hence a scalar||or , both correspond to the same element||where||where||1||1||1||1||1||1|
|Diagonalizable over , not over , eigenvalues square roots of||Square roots of||21||1||1||21|
|Not diagonal, has Jordan block of size two||(multiplicity 2) or (multiplicity 2). Each conjugacy class has one representative of each type.||where||where||24||2||2||48|
|Diagonalizable over , not over . Must necessarily have no repeated eigenvalues. Eigenvalues not square roots of .||Pair of conjugate elements of of norm 1, not square roots of -1. Each pair identified with its negative pair.||, irreducible; note that 's pair and 's pair get identified.||Same as characteristic polynomial||42||1||42|
|Diagonalizable over with distinct (and hence mutually inverse) diagonal entries||where where are square roots of . Note that the representative pairs and get identified.||, again with identification.||, again with identification.||56||1||56|
Interpretation as general linear group of degree three over field:F2
Compare with element structure of general linear group of degree three#Conjugacy class structure. Since many cases (namely, those that rely on distinct eigenvalues) do not arise over field:F2, the cases have been omitted in the table below.
|Nature of conjugacy class||Eigenvalues||Characteristic polynomial||Minimal polynomial||Size of conjugacy class||Number of such conjugacy classes||Total number of elements||Semisimple?||Diagonalizable over ?|
|Diagonalizable over with equal diagonal entries, hence a scalar||1||1||1||Yes||Yes|
|Diagonalizable over , not over||Distinct Galois conjugate triple of elements in . If one of the elements is , the other two are and .||irreducible degree three polynomial over||same as characteristic polynomial||24||2||48||Yes||No|
|One eigenvalue is in , the other two are in||one element of , pair of Galois conjugates over in .||product of linear polynomial and irreducible degree two polynomial over||same as characteristic polynomial||56||1||56||Yes||No|
|Has Jordan blocks of sizes 2 and 1 with equal eigenvalues over||with||42||1||42||No||No|
|Has Jordan block of size 3||with||same as characteristic polynomial||21||1||21||No||No|