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**Sample text**

Prove each mapping is well-defined and apply the theorem appropriately. 2 Homomorphisms and Subgroups 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Hint(2/5): For the counter example consider the a homomorphism between the multiplicative monoids of Z3 and Z6 . 1 Homomorphisms . . . . . Abelian Automorphism . . . Quaternions . . . . . . D4 in R2×2 . . . . . . . Subgroups . . . . . . . Finite subgroups . . . . . nZ . . . . . . . . . Subgroups of Sn . . .

Also, (cac−1 )−1 = (c−1 )−1 a−1 c−1 = ca−1 c−1 , which leads to (cac−1 )n = (cac−1 )−(−n) = ((cac−1 )−1 )n = (ca−1 c−1 )n = ca−(−n) c−1 = can c−1 , and so in general (cac−1 )n = can c−1 for all n ∈ Z. If n is the finite order of a, then an = e and so (cac−1 )n = cec−1 = e; so |cac−1 | ≤ |a|. Equally important, when the order of cac−1 is m, e = (cac−1 )m = cam c−1 ; thus am = c−1 c = e, so |a| ≤ |cac−1 |. Once again the conditions implicitly imply that if one order is finite, then so is the other; therefore, |a| = |cac−1 | when either has finite order.

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 . . . . . . . . . . . . . . . . . . . . . . . .