Difference between revisions of "Math 360, Fall 2013, Assignment 11"
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==Theorems== |
==Theorems== |
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| + | #Classification of Transitive Actions<p>\(G\) is a group, \(X\) is a transitive \(G\)-set. Choose any \(x\in X\). Then:$$ |
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| + | X \simeq G/G^x |
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| + | $$</p> |
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==Book Problems== |
==Book Problems== |
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Revision as of 08:05, 17 November 2013
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:
- Action (of a group $G$ on a set $X$).
- $G$-set.
- Homomorphism (of $G$-sets).
- Isomorphism (of $G$-sets).
- Orbit (in a $G$-set).
- Transitive action.
- Isotropy group (of a point in a $G$-set).
- Ring.
- Homomorphism (of rings).
- Unity.
- Unit (warning: this is not a synonym for "unity").
- Division ring.
- Field.
- Subring.
- Subfield.
Carefully state the following theorems (you need not prove them):
- Classification of transitive actions (we stated this in class; in the book it appears only as Exercise 16.15).
Solve the following problems:
- Section 16, problems 2, 3, and 9.
- Section 18, problems 3, 5, 7, 8, 11, 12, 14, and 17.
Questions
Definitions
- 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:
- \(G\)-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:
- 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:
- Isomorphism of \(G\)-sets.
Definition:
An isomorphism between two \(G\)-sets is a bijective homomorphism.
Example:
The example above is an isomorphism.
Non-Example:
- 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 \(4\).
- 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:
- 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:
- 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:
Example:
The integers are a ring.
Non-Example:
- 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:
- 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:
- 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.
- 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:
- 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:
- Subring.
Definition:
Given a ring \(R\) and a subset of \(R\) called \(K\), \(K\) is a subring of \(R\) if:$$ 0 \in K\\ x,y \in K \rightarrow x+y \in K\\ x,y \in K \rightarrow xy \in K x \in K \rightarrow -x \in K $$
Example:
The integers are a subring of the rationals.
Non-Example:
- 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
- Classification of Transitive Actions
\(G\) is a group, \(X\) is a transitive \(G\)-set. Choose any \(x\in X\). Then:$$ X \simeq G/G^x $$
