Question #285571

Let G be a group of order 11 2 :1 32 . H ow m any 11 -sylow subgroups and 13 sylow subgroups are there in G ?

1
Expert's answer
2022-01-10T12:26:07-0500

Solution:

O(G)=112132O(G)=11^{2} 13^{2}

The number of 11 Sylow subgroups are of the form t=1+11 k

since t divides O(G)

t divides 1121321+11k divides 1321+11k=1k=0\Rightarrow t\ divides\ 11^{2} 13^{2} \\\Rightarrow 1+11 k\ divides\ 13^{2} \\\Rightarrow 1+11 k=1 \\\Rightarrow k=0

Number of 11- Sylow subgroup =1

Since all 11-Sylow subgroups are conjugate and there is only one 11-Sylow subgroup implies the 11-Sylow subgroup is normal.

With the similar argument we can show that there is one 13-Sylow subgroup which is normal.


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