By James Wilson

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D4 in R2×2 . . . . . . . Subgroups . . . . . . . Finite subgroups . . . . . nZ . . . . . . . . . Subgroups of Sn . . . . . Subgroups and Homomorphisms Z2 ⊕ Z2 lattice . . . . . . Center . . . . . . . . Generators . . . . . . . Cyclic Images . . . . . . Cyclic Groups of Order 4 . . . Automorphisms of Zn . . . . Generators of PruferGroup . . Join of Abelian Groups . . . Join of Groups . . . . . . Subgroup Lattices . .

47 47 48 48 48 49 49 51 51 52 Order of Elements. Let a, b be elements of a group G. Show that |a| = |a−1 |; |ab| = |ba|, and |a| = |cac−1 | for all c ∈ G. Proof: Consider the cyclic group generated by an element a. 3, |a| = |a−1 |. Suppose the order, n, of ab is finite, so that (ab)n = e. We re-associate the product as follows: (ab) · · · (ab) = a(ba) · · · (ba)b = a(ba)n−1 b.

Thus H ∨ K ⊆ HK. 8. Therefore HK = H ∨ K. Now suppose we have a finite collection of subgroups H1 , . . , Hn of G. Since multiplication is associative, H1 · · · Hn = (H1 · · · Hn−1 )Hn , as the elements a1 · · · an = (a1 · · · an−1 )an . Suppose H1 · · · Hn = H1 ∨ · · · ∨ Hn for some n ∈ Z+ . Then H1 · · · Hn+1 = (H1 · · · Hn )Hn+1 = (H1 ∨ · · · ∨ Hn )Hn+1 = n n (H1 ∨· · ·∨Hn )∨Hn+1 . Finally i=1 Hi ∨Hn+1 is defined as ( i=1 Hi )∪Hn+1 Which is simply H1 ∨ · · · ∨ Hn+1 . Therefore by induction, H1 ∨ · · · ∨ Hn = H1 · · · Hn , for all n ∈ Z+ .

### A Hungerford’s Algebra Solutions Manual by James Wilson

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