By I. M. Isaacs

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**Example text**

In order to ensure that this actually is an X-composition series, we must know that Nr−1 /D is X-simple, which is true again by the diamond lemma. Similarly, we may gain a new composition series for Ms−1 in a similar fashion, by appending Ms−1 /D to T . We shall call these new composition series Tr and Ts , respectively. Now our inductive hypothesis applied to R0 and Tr tells us that R0 ≡ Tr , so the length of Tr = r −1. Likewise, the length of Ts = 1 + the length of T = length of Tr = r − 1. We again use the inductive hypothesis to conclude that Ts ≡ S0 ; we may therefore conclude that s − 1 = r − 1 and therefore that s = r .

3. Let G be an external direct product Q1 × Q2 × . . × Qr where each Qi is a p-group for some prime p. Then G is nilpotent. Proof. 1. Also, we know that Qi ∼ = Qi , so each Qi is nilpotent. From our homework, we know that F(G), the Fitting subgroup, is the (unique) largest r Qi ⊆ F(G), so G = F(G). nilpotent subgroup of any group G. Thus G = i=1 Yet F(G) is nilpotent, so G must also be nilpotent. We now get to the apex of this section: the Fundamental Theorem of Finite Abelian Groups. 3. Let G be finite and abelian.

I therefore just write out the phrase “internal direct product” or insert 33 clauses like, “where this product is direct”. The notation is really clumsy and for that I apologize. 1. Suppose that G1 , G2 , . . Gr are groups, and let P = {(x1 , x2 , . . , xr )|xi ∈ Gi }. We can make P into a group by defining (x1 , x2 , . . , xr ) · (y1 , y2 , . . yr ) = (x1 y1 , x2 y2 , . . , xr yr ). When we do this, we say that P is the external direct product of the groups Gi , and we write P = G1 × G2 × .