Let G be a group of order 143 = 11 * 13, and, as usual, let Z (G) denote the center of G. Assume that we have found an element x ∈ Z (G) with x = e. What are the possibilities for the |Z (G) |? Prove any assertions you make.

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Tapia

Mathematics

17 June, 06:50

Let G be a group of order 143 = 11 * 13, and, as usual, let Z (G) denote the center of G. Assume that we have found an element x ∈ Z (G) with x = e. What are the possibilities for the |Z (G) |? Prove any assertions you make.

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Creedence · Answer 1

Step-by-step explanation:

Let G be a group of order 143.

By the Sylow Third Theorem, the number of Sylow 11-subgroups of G is 1+11k and is a factor of 13,

Then, k=0 and there is an element a∈G such that Ord (a) = 11 and ⟨a⟩ is normal to G.

Also, the number of Sylow 13-subgroups of G is 1+13k and is a factor of 11,

Then, k=0 and there is an element b∈G in such away that Ord (b) = 13 and ⟨b⟩ is normal to G.

Since ⟨a⟩∩⟨b⟩={e}, it was discovered that ab=ba, therefore Ord (ab) = 143.