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Exercise 4.3.1 Verify the claims in the proof of Theorem 4.3.2. Specifically, prove that (i) a - a (91,..., 9p) = (ga+1,..., ga+p) defines an action of Z, on GP and (ii) X = {(9₁,..., 9p) EGP | 91929p = e} is invariant by Zp and hence the Z, action on GP restricts to an action on X. Hint: for the last part, you need to show that for all a € Zp, 91 9p = e if and only if ga+1 ga+p = e.


Exercise 4.2.3 Edges e, e' of a tetrahedron T are said to be opposite if they are disjoint (that is, they do not share a vertex). The 6 edges can be partitioned into a set X of three pairs of opposite edges. Prove that G3, the group of symmetries of T, acts on X and the kernel K< G3 is a normal subgroup of order 4. The previous exercise is interesting in that it provides a geometric description of a normal subgroup of G3 S₁ different than A₁. It turns out that for any n 4, the only normal subgroup of S₁ is An (and {(1)} and Sn, of course).


Exercise 4.3.4 Let p≥ 2 be a prime and prove that any group of order p² is isomorphic to either Zp² or Zpx Zp. Hint: If there exists an element of order p², then GZp², so assume that this is not the case and deduce that every nonidentity element of G has order p. Then prove that for any two elements g, h E G, either (g) = (h) or (g) n(h) = {e}. Finally, prove that there are two nonidentity elements g, h E G so that (g) n(h) = {e}, and apply Proposition 3.7.1.


5. (14 points) Suppose that : G→ H is a group homomorphism, K <H is a subgroup, and ¹(K) = {g €G | (g) € K}. Prove that -¹(K) is a subgroup of G. 6. (14 points) Suppose A, B4G are normal subgroups such that AnB = {e}. Prove that if a EA and be B, then ab = ba. IMPORTANT: You should not simply quote the Proposition from class stating that AB Ax B. Instead, you should supply the first part of that proof of that proposition, stating that the elements of A commute with the elements of B.


Exercise 3 Let G be a group of order 52-72-19-23275. (a) Prove that G contains exactly one subgroup of order 49. Prove furthermore that if N <G with |N| then N is normal. (b) Prove that G/N is isomorphic to either Z19 × Z25 or Z19 × Z5 X Z Suggestion: Similar to the Exercise 2c, exept apply Proposition 3.7.1 instead of 3.7.7. 49 (c) Let Ps and P19 be Sylow 5- and 19-subgroups of G, respectively. Prove that NP5 and NP19 are both subgroups of G and that NPN X P5, and NP19 N x P19-


Exercise 4 Let G be a group of order 2³ - 7 = 56. Prove that G contains at least one normal subgroup. Suggestion: Prove that G contains either a normal Sylow 2-subgroup or a normal Sylow 7-subgroup. Do this by assuming otherwise and counting elements. Be careful with intersections when doing your count.


Let A CR, and let x Є R. We say that the point x is far away from the set A if and only if:Ed >0: No element of A belongs to the set [x − d, x].


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