a) Find the number of elements of order 5 in a simple group of order 120. b) Let G be a group of order 154=(2)(7)(11). 1) Show that G has a normal subgroup of order 77. 2) Find the exact number of the Sylow 7-subgroups of G. (explain)
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Let G be a group of order 2022 = 2 · 3 · 337. (a) Prove that G has a group unique subgroup of order 337, and that this subgroup is normal in G (b) Assuming that G has more than one element of order 3, how many elements of of order 3 will G have? [Hint: figure out how many Sylow 3-subgroups G has and go from there].
Let G = Z5 ⊕ Z10. (a) How many elements of order 5 in G? (b) Consider H = h(1, 2)i - a subgroup of G. Why H is a normal subgroup of G? And what is the order of the factor group G/H?
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