10-1100/Homework Assignment 3 - Revision history

Drorbn at 12:02, 15 November 2010

2010-11-15T12:02:45Z

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	'''Problem 3.''' (Selick) Show that the group <math>Q</math> of unit quaternions (<math>\{\pm 1, \pm i, \pm j, \pm k\}</math>, subject to <math>i^2=j^2=k^2=-1\in Z(Q)</math> and <math>ij=k</math>) is not a semi-direct product of two of its proper subgroups.		'''Problem 3.''' (Selick) Show that the group <math>Q</math> of unit quaternions (<math>\{\pm 1, \pm i, \pm j, \pm k\}</math>, subject to <math>i^2=j^2=k^2=-1\in Z(Q)</math> and <math>ij=k</math>) is not a semi-direct product of two of its proper subgroups.

	'''Problem 4.''' (Qualifying exam, September 2008) Let <math>G</math> be a group and <math>p</math> be a prime. Show that if <math>H</math> is a <math>p</math>-subgroup of <math>G</math>, then <math>(N_G(H):H)</math> is congruent to <math>(G:H)</math> mod <math>p</math>. You may wish to study the action of <math>H</math> on <math>G/H</math> by multiplication on the left.		'''Problem 4.''' (Qualifying exam, September 2008) Let <math>G</math> be a finite group and <math>p</math> be a prime. Show that if <math>H</math> is a <math>p</math>-subgroup of <math>G</math>, then <math>(N_G(H):H)</math> is congruent to <math>(G:H)</math> mod <math>p</math>. You may wish to study the action of <math>H</math> on <math>G/H</math> by multiplication on the left.

	'''Problem 5.''' (easy)		'''Problem 5.''' (easy)

Drorbn at 02:27, 3 November 2010

2010-11-03T02:27:16Z

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	{{In Preparation}}

	This assignment is due at class time on Tuesday, November 16, 2010.		This assignment is due at class time on Tuesday, November 16, 2010.

Drorbn at 23:46, 2 November 2010

2010-11-02T23:46:15Z

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	===Solve the following questions===		===Solve the following questions===

			'''Problem 1.''' (Selick) Show that any group of order 56 has a normal Sylow-<math>p</math> subgroup, for some prime <math>p</math> dividing 56.
	<center>'''Barring the unforeseen, will be posted by midnight, November 2, 2010.'''</center>

			'''Problem 2.''' (Qualifying exam, May 1997) Let <math>S_5</math> act on <math>({\mathbb Z/5})^5</math> by permuting the factors, and let <math>G</math> be the semi-direct product of <math>S_5</math> and <math>({\mathbb Z/5})^5</math>.
			# What is the order of <math>G</math>?
			# How many Sylow-5 subgroups does <math>G</math> have? Write down one of them.

			'''Problem 3.''' (Selick) Show that the group <math>Q</math> of unit quaternions (<math>\{\pm 1, \pm i, \pm j, \pm k\}</math>, subject to <math>i^2=j^2=k^2=-1\in Z(Q)</math> and <math>ij=k</math>) is not a semi-direct product of two of its proper subgroups.

			'''Problem 4.''' (Qualifying exam, September 2008) Let <math>G</math> be a group and <math>p</math> be a prime. Show that if <math>H</math> is a <math>p</math>-subgroup of <math>G</math>, then <math>(N_G(H):H)</math> is congruent to <math>(G:H)</math> mod <math>p</math>. You may wish to study the action of <math>H</math> on <math>G/H</math> by multiplication on the left.

			'''Problem 5.''' (easy)
			# Prove that in any ring, <math>(-1)^2=1</math>.
			# Prove that even in a ring without a unit, <math>(-a)^2=a^2</math>.
			(Feel free to do the second part first and then to substitute <math>a=1</math>).

			'''Problem 6.'''
			# (Qualifying exam, April 2009) Prove that a finite integral domain is a field.
			# (Qualifying exam, September 2008) Prove that in a finite commutative ring, every prime ideal is maximal.

			'''Problem 7.''' (Dummit and Foote) A ring <math>R</math> is called a ''Boolean ring'' if <math>a^2=a</math> for all <math>a\in R</math>.
			# Prove that every Boolean ring is commutative.
			# Prove that the only Boolean ring that is also an integral domain is <math>{\mathbb Z}/2</math>.

Drorbn: /* Solve the following questions */

2010-11-02T12:42:27Z

Solve the following questions

← Older revision		Revision as of 08:42, 2 November 2010
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	===Solve the following questions===		===Solve the following questions===

			<center>'''Barring the unforeseen, will be posted by midnight, November 2, 2010.'''</center>

Drorbn at 01:10, 2 November 2010

2010-11-02T01:10:05Z

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{{10-1100/Navigation}}
{{In Preparation}}

This assignment is due at class time on Tuesday, November 16, 2010.

===Solve the following questions===