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I am trying to understand the proof of the following lemma.

Lemma: The commutator subgroup of $F_{i+1}$ is a subgroup of the group $\langle x_1, \ldots, x_{i-1}\rangle = g^{-1}F_{i-1}g$, where $(1 \leq i \leq t-2)$.

Proof: Provided that $1 \leq i \leq t-2$, the groups $F_i = \langle x_0, \ldots, x_{i-1} \rangle$ and $g^{-1}F_ig = \langle x_1, \ldots, x_i \rangle$ are different, and they are both of index two in $F_{i+1}$, and consequently normal in $F_{i+1}$. Thus their intersection $\langle x_1,\ldots,x_{i-1} \rangle = g^{-1}F_{i-1}g$ is normal in $F_{i+1}$, and the quotient group $F_{i+1}/(g^{-1}F_{i-1}g)$ is Abelian, since it has order 4. Hence the commutator subgroup of $F_{i+1}$ is contained in $g^{-1}F_{i-1}g$.

The proof is from "Algebraic Graph Theory" by Biggs (1974), and is shown on page 123 (if anybody happens to have the book). Although the subject is algebraic graph theory, I am quite sure that the proof relies on regular group theory.

Edit: Here are some more details, which in retrospect were important

First, the group $F_i$ is defined as: $F_i = \langle x_0, x_1, \ldots, x_{i-1} \rangle$. There is the following relation between $g$ and $x_i$ $$x_i = g^{-i}x_0g^i$$

The elements $x_i$ are involutions, that is $x_i = x_i^{-1}$. The order $|F_i| = 2^i$. This (as mentioned by Jack, this gives immediately that the index of $F_i$ in $F_{i+1}$ is 2, since $|F_{i+1}/F_i| = |F_{i+1}|/|F_i|$. The group $F_0$ is the identity.

Current status: At this point, I am almost through the proof, and currently the only thing which I do not understand is the sentence:

Provided that $1 \leq i \leq t-2$, the groups $F_i = \langle x_0, \ldots, x_{i-1} \rangle$ and $g^{-1}F_ig = \langle x_1, \ldots, x_i \rangle$ are different

I do not see why this is true, and I do not see why it is important. If somebody can answer that sub-question, I will mark the answer as correct.

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Most of the proof requires knowing something about these groups $F_i$. However, the order 4 part is a little silly. If $N$ has index 2 in $G$, then $N$ is normal in $G$, and $G/N$ is abelian so $N$ contains the commutator subgroup. If $M$ is also a subgroup of index 2, then $M$ also contains the commutator subgroup. If both $M$ and $N$ contain something, then $M \cap N$ contains it too. –  Jack Schmidt Jun 13 '12 at 19:37
    
Some things that are false in general: I see no reason for $F_i$ to have index 2 in $F_{i+1}$, and there is no reason that $\langle X \rangle \cap \langle Y \rangle = \langle X \cap Y \rangle$. –  Jack Schmidt Jun 13 '12 at 19:39
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How are $F_i$ and $g$ defined ? –  Joel Cohen Jun 13 '12 at 19:39
    
I have added some additional information to the question. –  utdiscant Jun 13 '12 at 19:42
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Oh ho, knowing the order of $F_i$ is very good. That proves the index 2 thing immediately, and I suspect in this case the intersection thing. –  Jack Schmidt Jun 13 '12 at 20:01

1 Answer 1

up vote 1 down vote accepted

While this post has been here, I have figured out the last details of the proof. Below is my new expanded proof:

Provided that $1 \leq i \leq t-2$, the groups $F_i = \langle x_0, \ldots, x_{i-1} \rangle$ and $g^{-1}F_ig = \langle x_1, \ldots, x_i \rangle$ are different. This is true, since if they were equal, the group $F_i$ would include $x_i$, which would make it equal to $F_{i+1}$, but we know that the order of $F_i$ and $F_{i+1}$ is not the same.

Since $|F_i| = 2^i$, we have that $ |F_{i+1} / F_i| = \frac{|F_{i+1}|}{|F_i|} = \frac{2^{i+1}}{2^i} = 2 $ and the same holds for $g^{-1}F_ig$. Thus both $F_i$ and $g^{-1}F_ig$ are index 2 in $F_{i+1}$. Since a subgroup of index 2 is normal, we then have that $F_i$ and $g^{-1}F_ig$ are normal subgroups in $F_{i+1}$. Since the intersection of two normal subgroups is also a normal subgroup, this gives us, that their intersection $\langle x_1,\ldots,x_{i-1} \rangle = g^{-1}F_{i-1}g$ is normal in $F_{i+1}$. Again using the order of $F_i$, we see that the quotient group $F_{i+1}/(g^{-1}F_{i-1}g)$ has order 4, and is therefore Abelian. At last, we use that for a group $G$, the quotient group $G/N$ is Abelian, iff the commutator subgroup of $G$ is a subset of N. This gives that $F_{i+1}/(g^{-1}F_{i-1}g)$ is Abelian -- and shows that the commutator subgroup of $F_{i+1}$ is contained in $g^{-1}F_{i-1}g$.

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