class that simple groups te groups. Every finite group can be "decomposed" into "atoms" in the following sense: (a) Show that for any finite group G, there exists a strictly descending chain of subgroups G=Go G₁2 G ₂ Z Gn-17 Gn= {1} such that G₁+1 ◄ G₁ and G₁/G₁+1 is simple for all i ≥ 0. Such a chain of subgroups is called a composition series for G, and the factors G₁/G₁+1 are called composition factors of G. (Hint: Induction on the order of G and use the Correspondence Theorem.) (Remark: The composition factors of G are the “atoms" which build up G. It can be proved that they only depend on G and not on the choice of composition series.) (b) Find a composition series for S4, and find its composition factors.

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