Subgroup of D_n: Proving <f> Not Normal

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In summary, the conversation discusses the non-normality of the subgroup <f> in relation to a reflection f. The goal is to show that for any element g in D-n, g(r^kf)g^-1 is not contained in <f>. The conversation also clarifies that the subgroup <f> is usually written as {e,f} and that proper sentences should be used for clarity.
  • #1
rapple
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Homework Statement



S.T <f> is not normal. where f is a reflection

Homework Equations


<f>={e,r^0 f, r^1f,r^2f,..}
WTS For any g in D-n, g(r^kf)g^-1 Not In <F>

The Attempt at a Solution


Elements of D-n are r^k, r^kf
For r^k, (r^k)(r^if)(r^-k) is in <f>.

So I am stuck
 
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  • #2
rapple said:
WTS For any g in D-n, g(r^kf)g^-1 Not In <F>

No. You want to show there exists such a g. Just find one.
 
  • #3
The subgroup generated by f, which is what <f> usually means, is just {e,f} with e the identity. What you wrote is in fact all of D_n. You probably should write in proper sentences so that everyone is sure what you mean (i.e. no abbreviations that might not be universally understood) - reading mathematics is hard at the best of times, so please help us out and make it as easy as possible.
 

What is a subgroup of Dn?

A subgroup of Dn is a subset of the group Dn that is also a group under the same operation. It contains the identity element, and every element in the subgroup has an inverse within the subgroup.

What does it mean for a subgroup to be normal?

A subgroup is normal if it is invariant under conjugation by elements of the larger group. This means that for any element g in the larger group and any element h in the subgroup, the element ghg-1 is also in the subgroup.

How do you prove that a subgroup is not normal?

To prove that a subgroup is not normal, you can show that there exists an element in the larger group that, when conjugated with an element in the subgroup, does not result in an element within the subgroup. This is a counterexample that disproves the subgroup's normality.

What is the role of the generating element f in proving a subgroup is not normal?

The generating element f is used to construct the subgroup in question. It is often chosen as a convenient and easily identifiable element that can generate the entire subgroup. This allows for a simpler and more straightforward proof of the subgroup's normality or lack thereof.

Can a subgroup of Dn be normal for all values of n?

No, a subgroup of Dn can only be normal for certain values of n. For example, it can be shown that all subgroups of D3 are normal, but this is not the case for all values of n. For instance, subgroups of D4 are not always normal.

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