Brian Bi
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Section 2.4. Cyclic Groups

Exercise 2.4.1 Since $$a^7 = 1$$, it follows that $$ab = a^{15}b$$. We can use $$a^3 b = ba^3$$ to exchange $$b$$ with 3 factors of $$a$$; doing this 5 times; we obtain $$a^{15}b = ba^{15}$$. So $$ab = ba^{15} = ba$$.

Exercise 2.4.2

1. The $$n$$th roots of unity are the $$n$$ complex numbers $$e^{2\pi ik/n}$$ with $$k = 0, 1, \ldots, n-1$$. It is easily verified that this set contains 1 and is closed under multiplication and inverses, so it is a group. It is cyclic since it is generated by the element $$e^{2\pi i/n}$$.
2. If $$n$$ is odd, then the $$n$$th roots of unity occur in $$(n-1)/2$$ complex conjugate pairs together with 1, so the product is 1. If $$n$$ is even, then there are $$n/2-1$$ complex conjugate pairs together with 1 and -1, so the product is -1. In general, the product is $$(-1)^{n+1}$$.

Exercise 2.4.3 Let $$n$$ be the order of $$ab$$. Then $$1 = (ab)^n = a(ba)^{n-1}b$$. Left-multiplying by $$b$$ right-multiplying by $$b^{-1}$$ yields $$1 = ba(ba)^{n-1} = (ba)^n$$. Therefore the order of $$ba$$ is at most $$n$$. By exchanging the roles of $$a$$ and $$b$$ in the above argument, we see that the order of $$ab$$ is also at most the order of $$ba$$. Therefore the two orders are equal.

Exercise 2.4.4 Suppose $$G$$ is a group that contains no proper subgroup. If $$G$$ is nontrivial, then $$G$$ must be cyclic; otherwise the subgroup generated by some element of $$G$$ other than the identity would be a proper subgroup. If $$G$$ is cyclic and finite with order greater than 1, the order must be prime, since $$C_p$$ is a proper subgroup of $$C_{pq}$$ whenever $$p, q > 1$$. If $$G$$ is infinite cyclic, then it has proper subgroups, such as the subgroup generated by $$1 + 1$$. Thus, the possible groups are the trivial group and the cyclic groups of prime order.

Exercise 2.4.5 Let $$G$$ be a cyclic group and $$H$$ a subgroup of $$G$$. Let $$x$$ be a generator of $$G$$, and write $$H = \{x^i \mid i \in S\}$$ where the set $$S$$ is possibly infinite. Let $$d$$ be the GCD of $$S$$, which is defined even if $$S$$ is infinite; see Exercise 2.3.3. By definition, $$d$$ is a linear combination of the elements of $$S$$, say, $$d = \sum_{i \in S} c_i i$$ where all $$c_i$$'s are integers and only finitely many are nonzero; and $$d$$ generates the set of such linear combinations. This implies that $$x^d = \prod_{i \in S} (x^i)^{c_i}$$, so $$H$$ contains $$x^d$$, and that $$d$$ divides all $$i \in S$$ so that all $$x^i$$'s are powers of $$x^d$$. It follows that $$H$$ is cyclic with generator $$x^d$$.

Exercise 2.4.6 In general, let $$G = C_n$$ and let $$x$$ be some generator of $$C_n$$. Observe that $$m$$ and $$n$$ are relatively prime if and only if $$m$$ and $$-n$$ are relatively prime, so by Corollary 2.3.6, there exist $$r, s \in \mathbb{Z}$$ such that $$rm - sn = 1$$ if and only if $$m$$ and $$n$$ are relatively prime. If such $$r, s$$ exist, then $$(x^m)^r = x^{sn + 1} = x$$, so $$x^m$$ generates $$C_n$$. Conversely, if $$x \in \langle x^m \rangle$$, then there is some $$r$$ with $$x = (x^m)^r = x^{mr}$$, so $$x^{mr-1} = 1$$, implying $$mr - 1 = sn$$ for some $$s$$, or $$rm - sn = 1$$; so when $$m$$ is not relatively prime to $$n$$, $$x \notin \langle x^m \rangle$$. Therefore the number of elements of $$C_n$$ that generate $$C_n$$ is precisely $$\varphi(n)$$, the number of integers from 0 to $$n-1$$, inclusive, that are relatively prime to $$n$$. Therefore $$C_6$$ has 2 generators, $$C_5$$ has 4, and $$C_8$$ has 4.

Exercise 2.4.7 Since all non-identity elements of $$H$$ have order 2, each such element is its own inverse, so each element of $$H$$ has an inverse in $$H$$. It remains to verify that $$H$$ is closed under multiplication. We only need to establish that $$xy, x(xy), yx, y(xy), (xy)x$$, and $$(xy)y$$ are in $$H$$; the products with 1 are obviously in $$H$$, and the squares of elements in $$H$$ have already been discussed. Now $$x(xy) = (xx)y = 1y = y \in H$$. Similar reasoning shows that $$(xy)y \in H$$. And $$(yx)xy = y(xx)y = yy = 1$$, so $$yx$$ is the inverse of $$xy$$, implying that $$yx = xy \in H$$. This then allows us to establish that $$y(xy) = y(yx) = (yy)x = x \in H$$ and likewise $$(xy)x = (yx)x = y(xx) = y \in H$$.

Exercise 2.4.8 I can't see how solving (a) first makes (b) easier, so instead I will give a standalone solution to (b) and reduce (a) to (b).

1. Suppose $$A \in GL_n(\mathbb{R})$$ is given. Let $$c = \det A$$. Then, $$A = BC$$ where $$B = \diag(c, 1, 1, \ldots, 1)$$ and $$C$$ is identical to $$A$$ except that each entry in the first row is equal to $$1/c$$ times the corresponding entry of $$A$$. By part (b), $$C$$ can be written as a product of elementary matrices of the first type. $$B$$ is an elementary matrix of the third type. Therefore $$A$$ is a product of elementary matrices of the first and third types.
2. First, some notation. Let $$E_n(r, c, a)$$ denote the $$n \times n$$ elementary matrix of the first type with the entry $$a$$ located at position $$(r, c)$$. Let $$S_n(i_1, i_2, c)$$ denote the diagonal matrix with the entry $$c$$ at $$(i_1, i_1)$$, the entry $$1/c$$ at $$(i_2, i_2)$$, and ones elsewhere on the diagonal. Note that left-multiplication by $$S_n(i_1, i_2, c)$$ has the effect of scaling rows $$i_1$$ and $$i_2$$ by a factor of $$c$$ and $$1/c$$, respectively, while leaving other rows unchanged. Let $$T_n(i_1, i_2)$$ denote the matrix whose entries are as follows: \begin{equation*} T_n(i_1, i_2)_{r,c} = \begin{cases} 1 & \text{if $r = c$, and $r \neq i_1$, and $r \neq i_2$} \\ -1 & \text{if $r = i_1$ and $c = i_2$} \\ 1 & \text{if $r = i_2$ and $c = i_1$} \\ 0 & \text{otherwise} \end{cases} \end{equation*} Note that left-multiplication by $$T_n(i_1, i_2)$$ has the effect of swapping rows $$i_1$$ and $$i_2$$, then changing the sign of the new row $$i_2$$.

As the text suggests, we first consider the case $$n = 2$$. If one can work out how to express $$S_2$$ and $$T_2$$ matrices in terms of $$E_2$$ matrices, the idea can be extended to arbitrary $$n$$. Here one can easily verify that \begin{align*} S_2(1, 2, c) &= \begin{pmatrix} c & 0 \\ 0 & 1/c \end{pmatrix} \\ &= \begin{pmatrix} 1 & 0 \\ -1/c & 1 \end{pmatrix} \begin{pmatrix} 1 & c-1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & \frac{1-c}{c} \\ 0 & 1 \end{pmatrix} \\ &= E_2(2, 1, -1/c)\, E_2(1, 2,c-1)\, E_2(2, 1, 1)\, E_2(1, 2, (1-c)/c) \\ T_2(1, 2) &= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \\ &= \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} \\ &= E_2(1, 2, -1)\, E_2(2, 1, 1)\, E_2(1, 2, -1) \end{align*} This allows us to deduce a way of writing the $$S_n$$ and $$T_n$$ matrices as products of $$E_n$$ matrices in general: \begin{align*} S_n(i_1, i_2, c) &= E_n(i_2, i_1, -1/c)\, E_n(i_1, i_2, c-1)\, E_n(i_2, i_1, 1) E_n(i_1, i_2, (1-c)/c) \\ T_n(i_1, i_2) &= E_n(i_1, i_2, -1)\, E_n(i_2, i_1, 1)\, E_n(i_1, i_2, -1) \end{align*}

Now in general the idea is to use row reduction to write an arbitrary $$A \in SL_n(\mathbb{R})$$ as the product of $$E_n$$'s. The $$S_n$$ and $$T_n$$ matrices correspond naturally to certain operations done during row reduction. We claim that if the first $$k$$ columns of $$A$$ are the same as those of the $$n \times n$$ identity matrix, then $$A$$ can be written as a product of $$E_n$$'s. The result will be proven by induction on $$n - k$$.

Base case: If $$n - k = 1$$, then by expansion in minors, $$\det A = A_{n,n}$$, therefore $$A_{n,n} = 1$$. Therefore \begin{equation*} A = E_n(1,n,A_{1,n})\, E_n(2,n,A_{2,n}) \ldots E_n(n-1,n,A_{n-1,n}) \end{equation*}

Inductive case: Suppose $$n - k \ge 2$$ and the left $$k$$ columns of $$A$$ are the same as those of an identity matrix. We now work on column $$k+1$$. If $$A_{k+1,k+1}$$ is nonzero then let $$B = A$$. Otherwise, since $$A$$ has full rank, the $$(1+k)$$th column can't be a linear combination of the first $$k$$ columns, so there must be some $$m > k$$ with $$A_{m,k+1} \neq 0$$. In the process of row reduction, we would swap rows $$k+1$$ and $$m$$, but since we are working with matrices of determinant 1, we also have to change the sign of one of the rows. Here, this is accomplished by writing $$A = T_n(k+1, m) B$$. Note that the first $$k$$ columns of $$B$$ are also the same as those of the identity matrix, but $$B_{k+1,k+1} \neq 0$$. So we have reduced the problem of $$A$$ to that of $$B$$.

In the process of row reduction, we would now scale row $$k+1$$ by $$1/c$$ to make the entry at $$(k+1,k+1)$$ equal to 1. Since we are working with matrices of determinant 1, we have to scale some higher-numbered row by $$c$$ as well in order to preserve the determinant. Since $$n - k \ge 2$$, we can write $$B = S_n(k+1, k+2, B_{k+1,k+1})C$$ where the first $$k$$ rows of $$C$$ are the same as those of the identity matrix, but $$C_{k+1,k+1} = 1$$. So we have reduced the problem of $$B$$ to that of $$C$$.

The final step of row reduction is to zero out all elements in column $$k+1$$ other than the pivot. This is done using the $$E_n$$'s themselves (see (1.2.5) in the text). The result is that $$C = E_n(1, k+1, -C_{1,k+1})\, E_n(2, k+1, -C_{2,k+1})\ldots D$$ where the first $$k+1$$ columns of $$D$$ are the same as those of the identity matrix. By the inductive hypothesis, $$D$$ is a product of elementary matrices of the first kind. This completes the induction.

By taking $$k = 0$$, we recover the desired result in full generality.

Exercise 2.4.9 A permutation $$P$$ that equals its own inverse can be expressed as a product of disjoint transpositions, where each transposition $$(a\ b)$$ is formed from a pair of distinct elements with $$P(a) = b, P(b) = a$$. Conversely, every product of disjoint transpositions is involutive. So the number of elements of order 2 in $$S_4$$ is equal to the number of ways to select some disjoint pairs of elements from a set of size 4, not counting the empty set of pairs (corresponding to the identity, which has order 1). There are 6 permutations containing one transposition and 3 more containing two tranpositions, for a total of 9.

Exercise 2.4.10 Define $$A, B \in GL_2(\mathbb{R})$$ as follows: \begin{align*} A &= \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \\ B &= \begin{pmatrix} -1 & 0 \\ 1 & 1 \end{pmatrix} \end{align*} Then $$A^2 = B^2 = I$$, but \begin{align*} AB &= \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \\ (AB)^n &= \begin{pmatrix} 1 & 0 \\ n & 1 \end{pmatrix} \end{align*} so $$AB$$ does not have finite order. But in an abelian group, if $$a$$ has order $$m$$ and $$b$$ has order $$n$$, then $$(ab)^{mn} = (a^m)^n(b^n)^m = 1$$, so $$ab$$ has finite order too.

Exercise 2.4.11

1. Write $$P \in S_n$$ as a product of disjoint cycles. It suffices to show that the tranpositions generate the cycles. Suppose $$C = (a_1\ a_2 \ldots a_n)$$. Then it can be written as the product of transpositions $$C = (a_1\ a_2) (a_2\ a_3) \ldots (a_{n-1}\ a_n)$$, and we are done.
2. Write $$P \in A_n$$ as a product of transpositions $$P = \tau_1 \tau_2 \ldots \tau_{2k-1} \tau_{2k}$$. The number of these transpositions must be even; we can pair them up like so: $$P = (\tau_1\tau_2) (\tau_3\tau_4) \ldots (\tau_{2k-1}\tau_{2k})$$. It suffices to write each parenthesized factor as a product of 3-cycles. We can do so using the fact that if $$a, b, c, d$$ are distinct, then $$(a\ b)(c\ d) = (a\ d\ c)(a\ b\ c)$$; otherwise WLOG assume $$b = c$$, and then $$(a\ b)(c\ d) = (a\ b\ d)$$. This completes the proof. Note that we didn't use the fact that $$n \geq 3$$, so the result holds for $$n = 2$$ as well; in this case $$A_n$$ is trivial so it is generated by the (empty) set of 3-cycles.