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quester [9]
3 years ago
11

A group issimpleif it has no nontrivial proper normal subgroups. LetGbe a simple group of order168. How many elements of order 7

are there inG
Mathematics
1 answer:
stepladder [879]3 years ago
3 0

Answer:

6*8=48 groups with elements of order 7

Step-by-step explanation:

For this case the first step is discompose the number 168 in factors like this:

168 = 8*3*7= 2^3 *3*7

And for this case we can use the Sylow theorems, given by:

Let G a group of order p^{\alpha} m  where p is a prime number, with m\leq 1 and p not divide m then:

1) Syl (G) \neq \emptyset

2) All sylow p subgroups are conjugate in G

3) Any p subgroup of G is contained in a Sylow p subgroup

4) n(G) =1 mod p

Using these theorems we can see that 7 = 1 (mod7)

By the theorem we can't have on one Sylow 7 subgroup so then we need to have 8 of them.

Every each 2 subgroups intersect in a subgroup with a order that divides 7. And analyzing the intersection we can see that we can have 6 of these subgroups.

So then based on the information we can have 6*8=48 groups with elements of order 7 in G of size 168

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