The groups (51-55, 56-60, etc), also called class intervals, are of width 5 The midpoints are in the middle of each class: 53, 58, 63 and 68 We can estimate the Mean by using the midpoints .
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Let G be a simple group of order 56. There are 8 subgroups of order 7, consuming 8×6 = 48 distinct elements of order 7. This leaves just enough for the subgroup of order 8, which is required. This sylow subgroup is unique, hence normal in G. Order 132 The group of order 132 is similar.
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10. The Class of Subgroups of Order 20 and Index 6. 2 11. The Class of Subgroups of Order 24 and Index 5. 2 12. The Class of Subgroups of Order 60 and Index 2. 2 13. The Class of Subgroups of Order 120 and Index 1. 2 1. The Class of Subgroups of Order 1 and Index 120. 1. 1 , Normal, Orb 15. Contains 1 group, 1 class. dpy. Below 2a, 2b[15 ...
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subgroups H’ of order 49 whose order 7 or less elements is T’. Each such subgroup T’ has a unique element (7,7a), a=0,1,2,…6,,or T’ is <(0,7)>, so there are 8 subgroups of order 7, Thus there are 7·8=56 cyclic subgroups of order 49 in G. 4. Prove that a subgroup H of index 2 in a finite group G is normal. Also, show that the
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