Characters of Normal Subgroup of Index 2

In summary, the conversation discusses the use of notation and inner product in proving a result involving a normal subgroup N of a finite group H and a character Chi belonging to the irreducible characters of H. The speaker is trying to show that for a given conjugacy class K of H, the values of two characters Theta 1 and Theta 2, both belonging to the irreducible characters of N, are equal for each element in K. The conversation also mentions the use of Frobenius Reciprocity and the construction of an induced character to reach this result.
  • #1
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Hi There,

Ok, I'm new to this so I'm sorry if this is abit warbled!...

We have a normal subgroup N of a finite group H such that [H:N]=2
We have a chatacter Chi belonging to the irreducible characters of H, Irr(H) , which is zero on H\N.

I have already shown that Chi restricted to N = Theta 1 + Theta 2, Where
Theta 1 is not equal to Theta 2, and both Theta 1, and Theta 2 belong to Irr(N).

I am now trying to show that given a conjugacy class K of H which is also a conjugacy class of N, we get Theta 1 (n) = Theta 2 (n) for n a member of K.

Intuitively, I can see this is true, I have seen examples in the alternating group A6, the normal subgroup of index 2 in S6 fro example.

I would be really greatful for a push in the right direction...I have books: 'James and Liebeck' and 'Isaacs'... if you could even direct me to some relevant info?

Wow, think my notation is abit crap, sorry.
 
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  • #2
It is traditional (though not necessarily ideal) to use ascii-tex to write maths in electronic form if you're not going to use the LaTeX capabilities of this site.

Thus [itex]\theta_1[/itex] is written as theta_1 (or \theta_1).

_ means subscript and ^ means superscript.

Right. I don't know if you're supposed to do it this way but here's my idea.

Consider Ind_N^H(theta_1).
 
  • #3
Thanks for the notation tip, don't know how to use LaTeX so will try the other method.
Ok, I'm looking at Ind_N^H(theta_1), using Frobenius Reciprocity, I have <Ind_N^H(theta_1) , Chi> = <theta_1, Chi_N> (where < , > denotes inner product)

Since Chi_N = theta_1 + theta_2 we get <Ind_N^H(theta_1) , Chi> = <theta_1, theta_1 + theta_2>

now, is <theta_1, theta_1 + theta_2> = <theta_1, theta_1> + < theta_1, theta_2> ? and where can i go from here to get to my result...as in have theta_1(n) = theta_2(n) for each n belonging to the conjugacy class K
 
  • #4
It's an inner product - it is of course (conjugate) linear in each variable. You have possibly learned (and I was assuming you had) that irreducible characters form an orthonormal basis of the class functions. However, if you didn't know that <A+B,C>=<A,C>+<B,C> then I have perhaps made a mistake in where you've got to in learning the material.

To get the final result you just need to look at how to construct in induced character.
 

1. What is a normal subgroup of index 2?

A normal subgroup of index 2 is a subgroup of a larger group that has half the order or size of the larger group. It is also a subgroup that is invariant under conjugation by all elements of the larger group.

2. How can we identify if a subgroup is normal and has index 2?

A subgroup is normal and has index 2 if it satisfies two conditions: 1) it is a subgroup of the larger group, and 2) for every element in the larger group, its inverse is also in the subgroup. In other words, the subgroup is closed under conjugation by all elements of the larger group.

3. What are the properties of a normal subgroup of index 2?

A normal subgroup of index 2 has several properties, including: it is a subgroup of the larger group, it has half the order or size of the larger group, it is invariant under conjugation by all elements of the larger group, and it forms a coset decomposition of the larger group.

4. How do normal subgroups of index 2 relate to the concept of normal subgroups?

Normal subgroups of index 2 are a special case of normal subgroups, as they are subgroups that are invariant under conjugation by all elements of the larger group. However, they have the additional property of having an index of 2, which means they have half the order or size of the larger group.

5. Can a group have more than one normal subgroup of index 2?

Yes, a group can have multiple normal subgroups of index 2. In fact, if a group has at least one normal subgroup of index 2, then it is guaranteed to have at least one more, as the quotient group formed by dividing the larger group by the normal subgroup will also have index 2 and therefore have its own normal subgroups of index 2.

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