By Evans L., Thompson R.

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**Extra resources for Introduction to algebraic topology**

**Example text**

Z/6Z ∼ = Z/2Z ⊕ Z/3Z but Z/4Z ∼ = Z/2Z ⊕ Z/2Z. 2 (Uniqueness Theorem for Finitely Generated Abelian Groups). Let A be a finitely generated abelian group and suppose A∼ = Z/d1 Z ⊕ Z/d2 Z ⊕ . . Z/dk Z ⊕ M ∼ = Z/d1 Z ⊕ Z/d2 Z ⊕ . . Z/dk Z ⊕ M where M, M are free, and d1 |d2 | . . |dk , d1 |d2 | . . |dk are integers > 1. Then M and M have the same rank (so are isomorphic), k = k and di = di , i = 1, . . , k. As above, we assume this was proved for you (or will be proved for you) in an algebra course.

Dk are integers > 1. Then M and M have the same rank (so are isomorphic), k = k and di = di , i = 1, . . , k. As above, we assume this was proved for you (or will be proved for you) in an algebra course. An alternate approach to the uniqueness statement first divides t(A) up as a direct sum of its Sylow subgroups. Each Sylow group has order pN for some prime p, and if we break it up as a direct sum of cyclic subgroups, we can arrange the divisibility relations simply by writing the factors in increasing order.

26. Let p : X x ∈ X. The action of π(X, x) on p−1 (x) is transitive. Proof. Let x˜ and y˜ lie over x. By assumption, since p is a covering, ˜ : I → X ˜ ˜ starting at x˜ X is path connected, so there is a path h ˜ is a loop based at x, so and ending at y˜. The projected path p ◦ h it represents some element α ∈ π(X, x). From the above definition, y˜ = x˜α. Continuing with generalities, let G act on a set X. If x ∈ X, consider the set Gx = {g ∈ G | xg = x}. We leave it to the student to check that Gx is a subgroup of G.

### Introduction to algebraic topology by Evans L., Thompson R.

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