Explore BrainMass

Cyclic group

This content was STOLEN from BrainMass.com - View the original, and get the already-completed solution here!

Let X be a prime. Prove or disprove that is cyclic for each normal subgroup K.

See attached file for full problem description.

© BrainMass Inc. brainmass.com October 16, 2018, 6:59 pm ad1c9bdddf


Solution Preview

First, we should know the definition of Dp.
Dp=<a,b> is generated by two elements a and b, where
a^2=b^p=1, ab=b^(-1)a.
So K=<b> is a normal subgroup of Dp. Because aba^(-1)=b^(-1) is in K.
Now I claim that K is the unique normal subgroup of Dp if p>=3 is a prime.
We know, |Dp|=2p, |K|=p, so [Dp:K]=2 and Dp = K union aK. So each ...

Solution Summary

This is a proof regarding a cyclic group.

Similar Posting

Group Theory : Cayley Tables, Cyclic Groups and Isomorphisms

? I have the following cayley tables (which is in modulo 9)

determine the order of each element . Prove that G is a cyclic group.

? Let be the symmetric group of degree 3 together with composition of maps. Is G isomorphic to ? Justify your answer.

? Let p be a prime number and G a group of order with identity element e. let and be a subgroup of G. prove that U is cyclic.

View Full Posting Details