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o(N) = N for all automorphisms o of G. (The term characteristic was first applied by G. Frobenius in 1895.) Prove that every sub- group of a cyclic group is characteristic. 2. Prove that the center of a group is characteristic. 3. The commutator subgroup G' of a group G is the subgroup gener- ated by the set (x ly 'xy lx, y E G). (That is, every element of G' has the form a "a," . . . a,", where each a; has the form x y 'xy, each i; = +1, and k is any positive integer.) Prove that G' is a char- acteristic subgroup of G. (This subgroup was first introduced by G. A. Miller in 1898.) 4. Prove that the property of being a characteristic subgroup is transi- tive. That is, if N is a characteristic subgroup of K and K is a char- acteristic subgroup of G, then N is a characteristic subgroup of G. 5. Let G = Z, O Z, O Z, and let H be the subgroup of SL(3, Z,) consisting of a H = 0 a, b, c E Z3 O (See Exercise 48 in Chapter 2 for the definition of multiplication.) Determine the number of elements of each order in G and H. Are G and H isomorphic? (This exercise shows that two groups with the same number of elements of each order need not be isomorphic.)