Solutions to Problem Set 18

Math 345/504

3-28-2018

1. In the group $\integer_5
   \times \integer_4$ , the cosets of the subgroup $\langle (3, 0)
   \rangle$ are

$$\eqalign{ \langle (3, 0) \rangle & = \{(0, 0), (3, 0), (1, 0), (4, 0), (2, 0)\} \cr (0, 1) + \langle (3, 0) \rangle & = \{(0, 1), (3, 1), (1, 1), (4, 1), (2, 1)\} \cr (0, 2) + \langle (3, 0) \rangle & = \{(0, 2), (3, 2), (1, 2), (4, 2), (2, 2)\} \cr (0, 3) + \langle (3, 0) \rangle & = \{(0, 3), (3, 3), (1, 3), (4, 3), (2, 3)\} \cr}.$$

(a) Compute

$$[(0, 3) + \langle (3, 0) \rangle] + [(0, 3) + \langle (3, 0) \rangle] \quad\hbox{in the quotient group}\quad \dfrac{\integer_5 \times \integer_4}{\langle (3, 0) \rangle}.$$

(b) Compute

$$-[(0, 3) + \langle (3, 0) \rangle] \quad\hbox{in the quotient group}\quad \dfrac{\integer_5 \times \integer_4}{\langle (3, 0) \rangle}.$$

(c) Compute

$$2 \cdot [(0, 2) + \langle (3, 0) \rangle] \quad\hbox{in the quotient group}\quad \dfrac{\integer_5 \times \integer_4}{\langle (3, 0) \rangle}.$$

(d) Make a table showing the order of each of the 4 cosets in the quotient group $\dfrac{\integer_5
   \times \integer_4}{\langle (3, 0) \rangle}$ .

Use your table to determine whether $\dfrac{\integer_5 \times \integer_4}{\langle (3, 0) \rangle}$ is isomorphic to $\integer_4$ or to $\integer_2 \times \integer_2$ .

(a)

$$[(0, 3) + \langle (3, 0) \rangle] + [(0, 3) + \langle (3, 0) \rangle] = (0, 2) + \langle (3, 0) \rangle.\quad\halmos$$

(b)

$$-[(0, 3) + \langle (3, 0) \rangle] = (0, 1) + \langle (3, 0) \rangle.\quad\halmos$$

(c) Compute

$$2 \cdot [(0, 2) + \langle (3, 0) \rangle] = \langle (3, 0) \rangle.\quad\halmos$$

(d)

$$\matrix{ \hbox{coset} & \hbox{order} \cr \noalign{\vskip2pt \hrule \vskip2pt} \langle (3, 0) \rangle & 1 \cr (0, 1) + \langle (3, 0) \rangle & 4 \cr (0, 2) + \langle (3, 0) \rangle & 2 \cr (0, 3) + \langle (3, 0) \rangle & 4 \cr}$$

Since there are elements of order 4, the quotient group is isomorphic to $\integer_4$ .


2. The group $U_{40}$ is

$$U_{40} = \{1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39\}.$$

The cosets of the subgroup $\langle 3 \rangle$ are

$$\eqalign{ \langle 3 \rangle & = \{1, 3, 9, 27\}, \cr 7 \cdot \langle 3 \rangle & = \{7, 21, 23, 29\}, \cr 11 \cdot \langle 3 \rangle & = \{11, 17, 19, 33\}, \cr 13 \cdot \langle 3 \rangle & = \{13, 31, 37, 39\}. \cr}$$

(a) Compute

$$\{11, 17, 19, 33\} \cdot \{7, 21, 23, 29\} \quad\hbox{in the quotient group}\quad \dfrac{U_{40}}{\langle 3 \rangle}.$$

(b) Compute

$$\{13, 31, 37, 39\}^2 \quad\hbox{in the quotient group}\quad \dfrac{U_{40}}{\langle 3 \rangle}.$$

(c) Compute

$$\{7, 21, 23, 29\}^{-1} \quad\hbox{in the quotient group}\quad \dfrac{U_{40}}{\langle 3 \rangle}.$$

(d) Make a table showing the order of each of the 4 cosets in the quotient group $\dfrac{U_{40}}{\langle 3 \rangle}$ .

Use your table to determine whether $\dfrac{U_{40}}{\langle 3 \rangle}$ is isomorphic to $\integer_4$ or to $\integer_2 \times \integer_2$ .

(a)

$$\{11, 17, 19, 33\} \cdot \{7, 21, 23, 29\} = \{13, 31, 37, 39\}.\quad\halmos$$

(b)

$$\{13, 31, 37, 39\}^2 = \{1, 3, 9, 27\}.\quad\halmos$$

(c)

$$\{7, 21, 23, 29\}^{-1} = \{7, 21, 23, 29\}.\quad\halmos$$

(d)

$$\matrix{ \hbox{coset} & \hbox{order} \cr \noalign{\vskip2pt \hrule \vskip2pt} \{1, 3, 9, 27\} & 1 \cr \{7, 21, 23, 29\} & 2 \cr \{11, 17, 19, 33\} & 2 \cr \{13, 31, 37, 39\} & 2 \cr}$$

Since every non-identity element has order 2, the quotient group is $\integer_2 \times \integer_2$ .


3. $GL(2, \real)$ is the group of $2 \times 2$ real matrices under matrix multiplication. Let

$$H = \left\{\left[\matrix{1 & x \cr 0 & 1 \cr}\right] \Bigm| x \in \real\right\}.$$

Prove that H is not a normal subgroup of $GL(2, \real)$ by specific counterexample.

$$\left[\matrix{ 2 & 1 \cr 1 & 1 \cr}\right] \left[\matrix{ 1 & 1 \cr 0 & 1 \cr}\right] \left[\matrix{ 2 & 1 \cr 1 & 1 \cr}\right]^{-1} =$$

$$\left[\matrix{ 2 & 1 \cr 1 & 1 \cr}\right] \left[\matrix{ 1 & 1 \cr 0 & 1 \cr}\right] \left[\matrix{ 1 & -1 \cr -1 & 2 \cr}\right] = \left[\matrix{ -1 & 4 \cr -1 & 3 \cr}\right] \notin H.\quad\halmos$$


[Math 504]

4. Two of the sporadic simple groups are known as the Monster Group and the Baby Monster Group. Find out their orders, and give a brief (one-sentence) description of the Monster. (That is, describe in general terms how it was constructed.)

The Monster is a sporadic simple group of order

$$2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71 \quad\hbox{or}\quad$$

$$\hskip0.5in 808017424794512875886459904961710757005754368000000000.$$

It was constructed in 1982 by Robert Griess, and is a group of rotations in 196883-dimensional space.

The Baby Monster is a sporadic simple group of order

$$2^{41} \cdot 3^{13} \cdot 5^6 \cdot 7^2 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 31 \cdot 47 = 4154781481226426191177580544000000.\quad\halmos$$


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