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.

Questions

For 16.9.c, the solutions manual says there are no isomorphic sub \(D_4\)-sets. I took sub-\(D_4\)-set to mean a \(G\)-set of \(D_4\) or any of its subgroups. Going by that definition, there are isomorphic sub-\(D_4\)-sets, because everything is isomorphic under the action of the trivial group. Is that right?

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

1. 16.3

   Orbits: (\(G = D_4\).)$$ \begin{eqnarray*} G1,G2,G3,G4 &=& \{1,2,3,4\}\\ Gs_1,Gs_2,Gs_3,Gs_4 &=& \{s_1,s_2,s_3,s_4\}\\ Gm_1,Gm_2 &=& \{m_1,m_2\}\\ Gd_1,Gd_2 &=& \{d_1,d_2\}\\ GC &=& \{C\}\\ GP_1,GP_2,GP_3,GP_4 &=& \{P_1,P_2,P_3,P_4\}\\ \end{eqnarray*} $$

1. 16.9

   a.The \(P_i\) and \(s_i\) orbits are isomorphic, given \(\phi(P_i) = s_i\).

   b. The \(s_i\) and \(i\) orbits can't be isomorphic, because they behave differently under reflections. On these two orbits, \(D_4\) is basically a subgroup of \(S_4\). The \(i\) orbit is basically \(\langle(1,2,3,4),(4,3,2,1)\rangle)\), while the \(s_i\) orbit is \(\langle(1,2,3,4),(1,3),(2,4)\rangle\). It might be better to say that the reflections are 4-cycles on \(i\) and transpositions on \(s_i\).

   c. Not sure. See question above.

1. 18.3

   \(11\cdot_{\mathbb{Z}_{15}} -4 = 1\)

2. 18.5

   \((2,3)\cdot_{\mathbb{Z}_5\times\mathbb{Z}_9}=(1,6)\)

3. 18.7

   \(n\mathbb{Z}\) is well-defined. This is a subset of \(\mathbb{Z}\), so we know all the properties of operations carry over. We just need to check for closure. Well, we know it's closed under addition, because of results from group theory. If \(x\) and \(y\) are \in \(\mathbb{Z}\), then \(x = np\) and \(y=nq\), so \(xy = npnq = n(pnq))\), so \(xy \in n\mathbb{Z}\). (Remember that \(n\mathbb{Z}\) is the set of all multiples of \(n\) in \(\mathbb{Z}\)). This subring is commutative, because integer multiplication is commutative. It does not contain unity unless \(n=\pm 1\). It is never a field.

4. 18.8

   \(\mathbb{Z}_+\) is not a ring, because it has no additive inverses. It's not even a group with respect to addition. (It is a group with respect to multiplication.)

5. 18.11

   The set \(\{a+\sqrt{2}b|a,b\in \mathbb{Z}\}\) is a ring. This is basically the complex numbers, but using \(\sqrt{2}\) instead of \(i\), and integers. This is not a field. Commutative and has unity.

6. 18.12

   The set \(\{a+\sqrt{2}b|a,b\in \mathbb{Q}\}\) is a ring.. This is also like the complex numbers, but with rationals and \(\sqrt{2}\) instead of \(i\). This is also a field (so it's commutative and has unity as well).

7. 18.14

   The units of \(\mathbb{Z}\) are 1 and -1.

8. 18.17

   Every element of \(\mathbb{Q}\) is a unit, except 0.