Math 360, Fall 2013, Assignment 11

The moving power of mathematical invention is not reasoning but the imagination.

- Augustus de Morgan

Carefully define the following terms, then give one example and one non-example of each:

1. Action (of a group $G$ on a set $X$).
2. $G$-set.
3. Homomorphism (of $G$-sets).
4. Isomorphism (of $G$-sets).
5. Orbit (in a $G$-set).
6. Transitive action.
7. Isotropy group (of a point in a $G$-set).
8. Ring.
9. Homomorphism (of rings).
10. Unity.
11. Unit (warning: this is not a synonym for "unity").
12. Division ring.
13. Field.
14. Subring.
15. Subfield.

Carefully state the following theorems (you need not prove them):

1. Classification of transitive actions (we stated this in class; in the book it appears only as Exercise 16.15).

Solve the following problems:

1. Section 16, problems 2, 3, and 9.
2. Section 18, problems 3, 5, 7, 8, 11, 12, 14, and 17.

Definitions

1. Action of a Group $G$ on a set $X$.

   Definition:

   Given $G$ and $X$, an action of $G$ on $X$ is a function $\mu:G\times X \rightarrow X$. Instead of actually writing $\mu(g,x)$, we write $g\cdot x$ or $gx$. This function must have the following properties:

   $$e\cdot x = x\\ g_1\cdot(g_2\cdot x) = (g_1g_2)\cdot x$$

   Example:

   Any group whose elements are functions (permutation groups) can create an action on the underlying set. So if we take $S_3$ are our set, we define $\mu$ as function application, and $X = \{1,2,3\}$. The properties follow from the properties of functions.

   Non-Example:

1. $G$-set.

   Definition:

   Given $G$, $X$ and $\mu$ as above, $X$ is a $G$-set.

   Example:

   In the above example, $\{1,2,3\}$ is an $S_3$-set.

   Non-Example:

1. Homomorphism of $G$-sets.

   Definition:

   Given $G$ and two $G$-sets $X$ and $Y$, a homomorphism between $X$ and $Y$ is a function $\phi:X\rightarrow Y$ such that:

   $$\phi(g\cdot x) = g\cdot\phi(x)$$

   Example:

   Let $G$ be the group of symmetries of $y=x$ (so identity and reflection across the diagonal). Let $X = \{(0,0)\}$ and $Y=\{(1,1)\}$. Define $\phi:X\rightarrow Y$ as $\phi(x) = (1,1)$. This is a homomorphism and these two $G$-sets are homomorphic.

   Non-Example:

1. Isomorphism of $G$-sets.

   Definition:

   An isomorphism between two $G$-sets is a bijective homomorphism.

   Example:

   The example above is an isomorphism.

   Non-Example:

1. Orbit in a $G$-set.

   Definition:

   Pick a point $x\in X$. The orbit of $x$ is the set:

   $$Gx = \{gx|g\in G\}$$

   So it's the set of everything you can get from $x$. (Arguably the set "generated" by $x$).

   Example:

   Let $G = \langle (1,2,3)\rangle \subset S_4$. The orbit of $1$ is $\{1,2,3\}$.

   Non-Example:

   For the same structures, $\{1,2,3\}$ is not the orbit of $4$. $4$ is the orbit of $4$.

1. Transitive Action.

   Definition:

   An action of $G$ on $X$ is transitive if their is only one orbit - all elements of $X$ generate the same set. (Not necessarily all of $X$).

   Example:

   Pick any action you want, and a point $x$. Find the orbit of $x$, call it $Gx$. Restrict $X$ to $Gx$. Now you have a transitive action.

   Non-Example:

1. Isotropy Group of a Point in a $G$-set.

   Definition:

   Choose $x\in X$. The isotropy group of $x$ is the set:

   $$G^{x} = \{g|gx = x\}$$

   So it's the set of all elements that leave $x$ fixed. This is, as the name would suggest, a group (specifically a subgroup of $G$).

   Example:

   Non-Example:

1. Ring.

   Definition:

   A ring is a (ternary?) structure $(R,+,\cdot)$, where $R$ is a set and $+$ and $\cdot$ are binary operations on $R$. Additionally:

   • $(R,+)$ is an abelian group.
   • $\cdot$ is associative.
   • Right and left distributive laws hold, meaning:
   $$x\cdot(y+z) = xy + xz\\ (x+y)\cdot z = xz + yz$$

   Example:

   The integers are a ring.

   Non-Example:

1. Ring Homomorphism.

   Definition:

   Take two rings $R$ and $Q$. A homomorphism between them is a function $\phi:R\rightarrow Q$ such that:

   $$\phi(x+y) = \phi(x) + \phi(y)\\ \phi(xy)=\phi(x)\phi(y)$$

   Example:

   The projection mapping $\phi:\mathbb{Z} \rightarrow \mathbb{Z}_6$ is a ring homomorphism.

   Non-Example:

1. Unity.

   Definition:

   Given a ring $R$, "unity" is a multiplicative identity for the ring. (Remember that rings are not guaranteed to have multiplicative identities).

   Example:

   1 is unity for the integers.

   Non-Example:

1. Unit.

   Definition:

   In a ring with unity (a ring with a multiplicative identity) a unit is an element with a multiplicative inverse.

   Example:

   1 and -1 are the only units in the integers.

   Non-Example:

   5 is not a unit in the integers - 1/5 is its inverse, which is not an integer.

1. Division Ring.

   Definition:

   A division ring is a ring in which every element has an inverse. This is not the same as a field - multiplication is not necessarily commutative. Also called a skew field. If it is not commutative, it is a strictly skew field. So the real numbers are a division ring/ skew field, but not a strictly skew field.

   Example:

   The rational numbers are a division ring. $GL(n)$ is a strictly skew field. (Not sure if this gets a new name when talking about it as a ring).

   Non-Example:

1. Field.

   Definition:

   A field is a commutative division ring (a ring with a multiplicative identity, multiplicative inverses, and multiplication is commutative.

   Example:

   The rational numbers are a field.

   Non-Example:

1. Subring.

   Definition:

   Given a ring $R$ and a subset of $R$ called $K$, $K$ is a subring of $R$ if:

   $$0 \in K\\ x,y \in K \rightarrow x+y \in K\\ x,y \in K \rightarrow xy \in K x \in K \rightarrow -x \in K$$

   So $K$ is a subset that is also a ring with respect to the operations $+,\cdot$ restricted to $K$.

   Example:

   The integers are a subring of the rationals.

   Non-Example:

1. Subfield.

   Definition:

   Given a field $F$ and a subset of $F$ called $K$, $K$ is a subfield of $F$ if:

   $$0 \in K\\ 1 \in K\\ x,y \in K \rightarrow x+y \in K\\ x,y \in K \rightarrow xy \in K\\ x \in K \rightarrow -x \in K\\ x \in K \rightarrow 1/x \in K$$

   Example:

   $\mathbb{Q}$ is a subfield of $\mathbb{R}$.

   Non-Example:

Theorems

1. Classification of Transitive Actions

   $G$ is a group, $X$ is a transitive $G$-set. Choose any $x\in X$. Then:

   $$X \simeq G/G^x$$

Book Problems

1. 16.2

   Isotropy subgroups are:

   $$\begin{eqnarray*} G^1,G^3 &=& \{\rho_0,\delta_2\}\\ G^2,G^4 &=& \{\rho_0,\delta_1\}\\ G^{s_1},G^{s_3},G^{P_1},G^{P_3} &=& \{\rho_0,\mu_1\}\\ G^{s_2},G^{s_4},G^{P_2},G^{P_4} &=& \{\rho_0,\mu_2\}\\ G^{m_1},G^{m_2} &=& \{\rho_0,\rho_2\,\mu_1,\mu_2}\\ G^{d_1},G^{d_2} &=& \{\rho_0,\rho_2,\delta_1,\delta_2\}\\ C &=& D_4 \end{eqnarray*}$$