Math 361, Spring 2014, Assignment 9

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

1. Solvable group.
2. $G$-set (this and related definitions are included for review, and can be found in Section 16 of the text).
3. Orbit (of a point in a $G$-set).
4. Isotropy group (at a point of a $G$-set).
5. Fixed point (of a $G$-set).

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

1. Theorem relating the cardinality of an orbit to the index of the isotropy group (Theorem 16.16 in the text).
2. Counting lemma for groups of prime power order (Theorem 36.1 in the text).
3. Cauchy's Theorem.
4. Sylow's First Theorem.
5. Theorem on solvability of groups of prime power order.

Solve the following problems:

1. Section 15, problem 39 (this is much too long to be the quiz question, but the fact that $A_n$ is a simple group for $n\geq5$ is a big deal, and this exercise will guide you through a proof).
2. Prove that any group of order 162 is solvable. (More generally, prove that for any prime $p$ and any positive integer $n$, any group of order $2p^n$ is solvable.)

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Questions:

Will this homework be on Exam II? I was not sure since the exam is in one week and homeworks near the exams have been left off before.--Robert.Moray (talk) 00:27, 8 April 2014 (EDT)

Let's say that the exam covers up to (and including) Monday's lecture. Then this assignment should be included, but the next will not be. --Steven.Jackson (talk) 09:01, 8 April 2014 (EDT)

Will we be responsible for the review terms like isotropy group and $G$-set? --Robert.Moray (talk) 09:21, 8 April 2014 (EDT)

Yes. --Steven.Jackson (talk) 13:35, 10 April 2014 (EDT)

I am confused. Is this homework based on section 35 and 36 of the textbook and I am going to assume that those two will on Monday's exam? Also did we skipped section 33?

Yes to both. (We have postponed our study of finite fields until after the general machinery of splitting fields is in place, probably in late April.) --Steven.Jackson (talk) 13:35, 10 April 2014 (EDT)

I was wondering if someone could elaborate on the difference between "factors" and "composition factors." What I think I know now is that a subnormal series has factors, a composition series has composotion factors, a composition series is a saturated subnormal series, and given a finite group $G$, there exists an isomorphic composition series for each composition series of $G$. --Robert.Moray (talk) 20:35, 10 April 2014 (EDT)

Any subnormal series has factors. If the series happens to be saturated, we call it a "composition series" and call its factors "composition factors." The Jordan-Holder Theorem tells us that any two composition series for the same group have isomorphic factors, so it is reasonable to speak of "the" composition factors of a given group (provided that some composition series exists -- which will always be the case if the group is finite.). --Steven.Jackson (talk) 22:05, 13 April 2014 (EDT)

Solutions:

Definitions:

1. Solvable group.

   A finite group, $G$ is said to be solvable if its composition factors are abelian

   Example:

   $\mathbb{Z}_{12}$ has composition factors $\mathbb{Z}_2$,$\mathbb{Z}_2$,$\mathbb{Z}_3$, so $\mathbb{Z}_{12}$ is solvable.

2. $G$-set (this and related definitions are included for review, and can be found in Section 16 of the text).

   A group $G$ is said to act on a set $X$ if there is some function $\cdot: G\times X\rightarrow X$ such that $$e\cdot x = x, \forall x\in X$$ $$g_1\cdot\left(g_2\cdot x\right) = \left(g_1g_2\right)\cdot x, \forall g_1,g_2\in G,\forall x\in X$$

3. Orbit (of a point in a $G$-set).

   The orbit of $x\in X$ is $$Gx =\lbrace g\cdot x|g\in G\rbrace$$

4. Isotropy group (at a point of a $G$-set).

   If $x\in X$, the isotropy group at x is $$G^x=\lbrace g\in G|g\cdot x = x\rbrace$$

5. Fixed point (of a $G$-set).

   If $Gx=\lbrace x\rbrace$ we sat that $x$ is a fixed point. The set of fixed points is $X^G$

   Example:

   In $D_4$, the origin is a fixed point

Theorems:

1. Theorem relating the cardinality of an orbit to the index of the isotropy group (Theorem 16.16 in the text).

   Let $X$ be a $G$-set and let $x\in X$. Then $|Gx| =\left[G: Gx\right]$. If $|G|$ is finite, then $|Gx| =\dfrac{|G|}{|G^X|}$

2. Counting lemma for groups of prime power order (Theorem 36.1 in the text).

   If $|G|=p^n$, where $p$ is prime and $X$ is any finite $G$-set, then $|X|\equiv |X^G|\left(mod p\right)$.

3. Cauchy's Theorem.

   If p is prime and $p| |G|$ then $G$ has a subgroup of order p.

4. Sylow's First Theorem.

   Suppose $p$ is prime and $p^i$ is a divisor of $|G|$. Then $G$ has a subgroup of order $p^i$ and each subgroup of order $p^i$ is normal in some subgroup of order $p^{i+1}$(provided that $p^{i+1} | |G|$)

5. Theorem on solvability of groups of prime power order.

   If $|G|= p^n$ , where $p$ is prime, then the composition $n$ factors are $\mathbb{Z}_p, \mathbb{Z}_p, \mathbb{Z}_p, \cdots, \mathbb{Z}_p,$ in particular, $G$ must be solvable.