The Jordan–Hölder Theorem

Hence, once we have determined one composition series for a (say, finite)
group, then we have uniquely determined composition factors whichcanbe
thought of as the ways of breaking our original group down into simple
groups. This is analogous to a ‘prime factorisation’ for groups.
To recognise when we have a composition series, we need to be able
to recognise simple groups. We have already observed (Example 3.6) that
cyclic groups of prime order are simple. It is not hard to show that these
are all the abelian simple groups. The following was proved in MT4003.
Theorem 4.7 Let n 5. ThenthealternatinggroupA n is simple.
It is worth pointing out that much, much more is known. A mammoth
effort by hundreds of mathematicians from the 1950s to the 1980s succeeded
in classifying the finite simple groups. The complete proof runs to tens of
thousands of pages of extremely complicated mathematics. More work is
still being done to check, clarify and simplify the proof. Nevertheless, it is
generally accepted that this Classification is correct, though when relying
upon it a mathematician would normally state that he or she is doing so.
Theorem 4.8 (Classification of Finite Simple Groups) Let G be a finite
simple group. Then G is one of the following:
(i) acyclicgroupofprimeorder;
(ii) an alternating group A n where n 5;
(iii) one of sixteen infinite families of groups of Lie type;
(iv) one of twenty-six sporadic simple groups.
The groups of Lie type are essentially ‘matrix-like’ groups which preserve
geometric structures on vector spaces over finite fields. For example, the first
(and most easily described) family is
PSL n (q) =
SL n(q)
Z(SL n (q)) .
To make it, first construct the group GL n (q) ofinvertiblen × n matrices
with entries from the finite field F of order q. Next, take the subgroup of
matrices of determinant 1:
SL n (q) ={ A ∈ GL n (q) | det A =1},
the special linear group. Now,factorbythecentreofSL n (q), which consists
of all scalar matrices of determinant 1:
Z(SL n (q)) = { λI | λ ∈ F , λ n =1}.
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