By Randall R. Holmes

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**Sample text**

N) Since each such representation is completely determined by its bottom row (and different bottom rows correspond to different permutations), the number of permutations of {1, 2, . . , n} is simply the number of arrangements of these numbers. Thus |Sn | = n!. 2 Cycle Let n be a positive integer and let i1 , i2 , . . , ir be distinct integers with 1 ≤ ij ≤ n. Let σ be the element of Sn that satisfies σ(ij ) = ij+1 i1 j < r, j = r, and σ(k) = k for all k ∈ / {i1 , i2 , . . , ir }. This permutation is written σ = (i1 , i2 , .

If G is abelian, then every subgroup of G is normal. 4 Normalizer of a set Let G be a group and let S be a subset of G. The normalizer of S in G is the set NG (S) = {g ∈ G | S g = S}. NG (S) is a subgroup of G. Theorem. Let H be a subgroup of G. (i) H NG (H). (ii) H G if and only if NG (H) = G. Proof. Since H is a normal subgroup of itself, it follows that H ⊆ NG (H). The remaining parts of the theorem are immediate from the definitions. 5 Product of subgroup and normal subgroup Let G be a group.

The length of (Gi ) is the number of its nontrivial factors. 4), it is a composition series. 7, (Gi ) is a composition series of G if and only if Gi+1 is a maximal normal subgroup of Gi for each i. A group need not have a composition series (Z has none for example). However, if G is finite and nontrivial, then it has a composition series since, for instance, one can be constructed by starting with G0 = G and recursively choosing Gi+1 to be a maximal normal subgroup of Gi (this process necessarily ending with Gr = {e} for some r since the orders of the subgroups strictly decrease).

### Algebra I by Randall R. Holmes

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