Daily Exercises 2 (Ch 6-?)

## Exercises

6.01: Find an isomorphism from the group of integers under addition to the group of even integers under addition.

Daily 10/2: Suppose $G$ is a cyclic group of order $n$. Prove that $G$ is isomorphic to $Z_n$.

6.4: Show that $U(8)$ is not isomorphic to $U(10)$

6.05: Show that $U(8)$ is isomorphic to $U(12)$.

6.9: In the notation of Theorem 6.1, prove that $T_e$ is the identity and that $(T_g)^{-1} = T_{g^{-1}}$.

6.11: For inner automorphisms $\phi$$g, \phi$$h$, and $\phi$$gh, prove that \phi$$g$$\phi$$h$=$\phi$$gh. 6.14: Find Aut(\mathbb{Z}_6) 6.20: Suppose that \phi: Z50 -> Z50 is an automorphism with \phi(11) = 13. Determine a formula for \phi(x) 6.21: Prove property 1 of Theorem 6.3. Daily Assignment 10/7: Write up a careful proof that \phi(x)=e^x is an isomorphism from R to R^+ 7.2: Let H be as in Exercise 1. How many left cosets of H in S$$4$ are there? (Determine this without listing them.)

7.8: Suppose $a$ has order 15. Find all of the left cosets of $<a^5>$ in $<a>$.

7.10: Let $a$ and $b$ be nonidentity elements of different orders in a group $G$ of order 155. Prove that the only subgroup of $G$ that contains $a$ and $b$ is $G$ itself.

7.14: Suppose that $K$ is a proper subgroup of $H$ and $H$ is a proper subgroup of $G$. If $|K|$ = 42 and $|G|$ = 420, what are possible orders of $H$?

7.15 Let G be a group with $|G|=pq$, where p and q are prime. Prove that every proper subgroup of G is cyclic.

7.20: Suppose H and K are subgroups of a group G. If |H| = 12 and |K| = 35, find |H∩K|. Generalize.

7.21: Suppose that $H$ is a subgroup of $S_4$ and that $H$ contains $(12)$ and $234$. Prove that $H=S_4$.

7.27: Let $|G|=33$. What are the possible orders for the elements of $G$? Show that $G$ must have an element of order 3.

7.29: Can a group of order $55$ have exactly $20$ elements of order $11$? Give a reason for your answer.

7.35: Let $G$={(1), (12)(34), (1234)(56), (13)(24), (1432)(56), (56)(13), (14)(23), (24)(56)}
a. Find the stabilizer of 1 and the orbit of 1.
b. Find the stabilizer of 3 and the orbit of 3.
c. Find the stabilizer of 5 and the orbit of 5.

7.37: Prove that the eight-element set in the proof of Theorem 7.4 is a group.

7.44: Let G be the group of rotations of a plane about a point P in the plane. Thinking of G as a group of permutations of the plane, describe the oribit of a point Q in the plane. (This is the motivation for the name "orbit.")

7.45: Let $G$ be the rotation group of a cube. Label the faces of the cube 1 through 6, and let $H$ be the subgroup of elements of $G$ that carry face 1 to itself. If $\sigma$ is a rotation that carries face 2 to face 1, give a physical description of the coset $H\sigma$.

7.48: a&b
Calculate the orders of the following:
a. The group of rotations of a regular tetrahedron (a solid with four congruent equilateral triangles as faces)
b. The group of rotations of a regular octahedron (a solid with eight congruent equilateral triangles as faces)

7.48: c&d
Calculate the orders of the following:
c. The group of rotations of a regular dodecahedron (a solid with twelve congruent regular pentagons as faces)
d. The group of rotations of a regular icosahedron (a solid with 20 congruent equilateral triangles as faces)

12.2: The ring {0,2,4,6,8} under addition and multiplication modulo 10 has a unity. Find it.

29.1:
Determine the number of ways in which the four corners of a square can be colored with two colors. (It is permissable to use a single color on all four corners.)

29.2:
Determine the number of different necklaces that can be made using 13 white beads and three black beads.

29.3:
Determine the number of ways in which the vertices of an equilateral triangle can be colored with five colors so that at least two colors are used.