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Mark Rosinski, markrosi@purdue.edu Joseph Lam, lam5@purdue.edu Beichen Xiao, xiaob@purdue.edu | Mark Rosinski, markrosi@purdue.edu Joseph Lam, lam5@purdue.edu Beichen Xiao, xiaob@purdue.edu | ||
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− | |||
<br> | <br> | ||
− | = | + | == ''P-groups & Sylow Theorem'' == |
+ | |||
+ | ---- | ||
+ | |||
+ | <br> | ||
= P-groups = | = P-groups = | ||
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<br> | <br> | ||
− | All | + | All content from this section is available from [http://people.maths.ox.ac.uk/craven/docs/lectures/pgroups.pdf here] as well as further information and proofs of these propositions on Automorphism Groups of order p<sup>n</sup>. |
<br> | <br> | ||
− | == | + | == Classification of p-groups == |
− | ''' | + | '''1. Regular p-groups''' |
− | + | Philip Hall, known for his work on group theory, initiated the Regular p-groups. | |
− | + | ||
− | + | ||
− | + | *'''Definitions:''' | |
+ | **For every <math>a, b \in G</math> there exist <math> c \in [<a,b>,<a,b>]</math> such that <span class="texhtml">''a''<sup>''p''</sup>''b''<sup>''p''</sup> = (''a''''b'''''<b>)<sup>''p''</sup>''c''<sup>''p''</sup></b></span> | ||
+ | **For every <math>a, b \in G</math> there exist <math>c_i \in [<a,b>,<a,b>]</math> such that <math>a^p b^p = (ab)^p c^p _1 c^p _2 . . . c^p _i</math> | ||
+ | **For evert <math>a, b \in G</math> and every positive integer <span class="texhtml">''n''</span> there exist elements <math>c_i \in [<a,b>,<a,b>]</math> such that <math>a^q b^q = (ab)^q c^q _1 c^q _2 . . . c^q _i</math> where <span class="texhtml">''q'' = ''p''<sup>''n''</sup></span> | ||
− | <br> <br> | + | <br> |
+ | All content from this section can be found [http://www-history.mcs.st-and.ac.uk/Biographies/Hall.html here] and [http://books.google.com/books?id=oyxnWF9ssI8C&lpg=PR1&pg=PA176#v=onepage&q&f=false here] | ||
+ | <br> | ||
+ | |||
+ | '''2. Powerful p-groups''' | ||
+ | |||
+ | *'''Definition:''' | ||
+ | **<span class="texhtml">''G''</span> is powerful if <math>\mho _1 (G) \supseteq G'</math> | ||
+ | **Let <math> N \triangleleft G </math>, then <span class="texhtml">''N''</span> is powerfully embedded in <span class="texhtml">''G''</span> if <math>\mho _1 (N) \supseteq [N,G]</math>. <span class="texhtml">''G''</span> is powerful '''IFF''' it is powerfully embedded in itself. If <span class="texhtml">''N''</span> is powerfully embedded in <span class="texhtml">''G''</span>, then <span class="texhtml">''N''</span> is powerful, more importantly, if <math>N \subseteq H \subseteq G</math> and <math> \frac{H}{N} </math> is cyclic, then <math>\mho _1 (H) \supseteq \mho _1 (N) \supseteq [N,H] = H'</math>, so H is powerful | ||
+ | |||
+ | <br> | ||
+ | |||
+ | *'''Proposition:''' | ||
+ | **G is powerful '''IFF''' G is powerfully embedded in G | ||
+ | **If N is powerfully embedded in G, then N is powerful and G centralizes <math>\frac{N}{N^p}</math> | ||
+ | **If <math>K \triangleleft G</math> and <span class="texhtml">''N''</span> is powerfully embedded in G then <math>\frac{N \times K}{K}</math> is powerfully embedded in <math>\frac{G}{K}</math> | ||
+ | **If N is powerfully embedded in G and <math>x \in G</math> then <x>N is powerful | ||
+ | **If M is powerfully embedded in G and N is powerfully embedded in G, then <math>M \times N</math> is powerfully embedded in G | ||
+ | |||
+ | <br> | ||
+ | |||
+ | All content from this section is available [http://ac.els-cdn.com/0021869387902110/1-s2.0-0021869387902110-main.pdf?_tid=4313cd0e-5a7a-11e3-a87b-00000aacb360&acdnat=1385896871_c3db54f10a7b71fce86d33825dd8041e here] and [http://books.google.com/books?id=34khoLiyP_QC&lpg=PA113&ots=4mE5X4ovFU&dq=powerful%20p%20groups%20properties&pg=PA113#v=onepage&q=powerful%20p%20groups%20properties&f=false here] as well as further information and proofs on Powerful p-groups | ||
+ | |||
+ | <br> | ||
+ | |||
+ | '''3. Special Groups''' | ||
+ | |||
+ | *'''Definition:''' | ||
+ | **Let G be a finite p-group, then G is defined to be special if '''either''' G is elementary abelian '''or''' G is of class 2 and G' = Φ(G) = Z(G) is elementary abelian. If G is a non-abelian special group with |Z(G)| = p, then G is defined to be extraspecial. | ||
+ | **Dihedral group <math>D_{2n} = <x,y: x^{n} = y^{2} = 1, x^{y} = x^{\rightharpoondown1}> = <a,b: a^{2} = b^{2} = 1, (ab)^{n} = 1></math> | ||
+ | **Quaternion group <math>Q_{4n} = <x,y: x^{2n} = y^4 = 1, x^y = x^{\rightharpoondown1}, y^2 = x^n></math> | ||
+ | **Semidihedral group <math>SD_{2n} = <x,y: x^{2^{n-1}} = y^2 = 1, x^y = x^{2^{n-2}-1}></math> | ||
+ | **Modular p-group <math>Mod_n (p) = <x,y;: x^{p^{n-1}} = y^p = 1, x^y = x^{1+p^{n-2}} | ||
+ | </math> | ||
+ | |||
+ | |||
+ | <br> | ||
+ | All content from this section is available [http://people.maths.ox.ac.uk/craven/docs/lectures/pgroups.pdf here] | ||
+ | |||
+ | == <br> <br> == | ||
== Sylow's Theorems == | == Sylow's Theorems == | ||
+ | '''Who is Sylow?''' | ||
+ | |||
+ | Peter Ludwig Mejdell Sylow (12 December 1832 – 7 September 1918) was a Norwegian mathematician who proved foundational results in group theory. He was born and died in Christiania (now Oslo). | ||
+ | |||
+ | Sylow was a high school teacher in Halden, Norway, from 1858 to 1898, and a substitute lecturer at Christiania University in 1862, covering Galois theory. It was then that he posed the question that led to his theorems regarding Sylow subgroups. Sylow published the Sylow theorems in 1872, and subsequently devoted eight years of his life, with Sophus Lie, to the project of editing the mathematical works of his countryman, Niels Henrik Abel. | ||
+ | |||
+ | He was appointed professor of Christiania University in 1898. | ||
− | '''Notation:''' | + | <br> '''Notation:''' |
Syl<sub>p</sub>(G) = the set of Sylow p-subgroups of G | Syl<sub>p</sub>(G) = the set of Sylow p-subgroups of G | ||
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All content from this section and proofs of these Theorems can be found [http://math.berkeley.edu/~sikimeti/SylowNotes.pdf here] | All content from this section and proofs of these Theorems can be found [http://math.berkeley.edu/~sikimeti/SylowNotes.pdf here] | ||
+ | <br> '''Application of Sylow theorems:''' | ||
− | + | *In D<sub>6</sub> all reflections are conjugate, as reflections correspond to Sylow 2-subgroups. | |
− | + | ||
− | + | [[Image:Application.png]] | |
+ | <br> | ||
<br> | <br> | ||
+ | |||
+ | *In D<sub>12</sub> reflections no longer correspond to Sylow 2-subgroups, and fall into two conjugacy classes. | ||
+ | |||
+ | [[Image:Application2.png]] | ||
+ | |||
+ | <br> <br> | ||
<br> | <br> | ||
Line 127: | Line 183: | ||
== References == | == References == | ||
− | + | Alavaro Loustau. ''Latex Tutorial.'' Retrieved from [http://omega.albany.edu:8008/Symbols.html omega.albany.edu:8008/Symbols.html ]<br> | |
+ | |||
+ | Alexander Lubotzky. "Powerful p-Groups. I. Finite Groups." Retrieved from [http://ac.els-cdn.com/0021869387902110/1-s2.0-0021869387902110-main.pdf?_tid=ca85355c-5a4d-11e3-a92f-00000aab0f6c&acdnat=1385877770_8465d8f5d1271c7df935737fe18a6211 0021869387902110/1-s2.0-0021869387902110-main.pdf] | ||
+ | |||
+ | Bruno Kahn. "A Characterization of Powerfully Embedded Normal Subgroups of a p-Group." Retrieved from [http://ac.els-cdn.com/S0021869396968465/1-s2.0-S0021869396968465-main.pdf?_tid=84c3502e-5a4f-11e3-af52-00000aab0f01&acdnat=1385878513_982cb60bacea08c6f2f7ce1ef2b29f6c S0021869396968465/1-s2.0-S0021869396968465-main.pdf] | ||
+ | |||
+ | Charles Richard Leedham-Green. "The Structure of Groups of Prime Power Order." Retrieved from [http://books.google.com/books?id=34khoLiyP_QC&lpg=PA113&ots=4mE5X4ovFU&dq=powerful%20p%20groups%20properties&pg=PA113#v=onepage&q&f=false id=34khoLiyP_QC&lpg=PA113&ots=4mE5X4ovFU&dq] | ||
+ | |||
+ | David A. Craven. ''The Theory of p-Groups. ''Retrieved from [http://people.maths.ox.ac.uk/craven/docs/lectures/pgroups.pdf craven/docs/lectures/pgroups.pdf] | ||
+ | |||
+ | J J O'Connor. ''Philip Hall.'' Retrieved from [http://www-history.mcs.st-and.ac.uk/Biographies/Hall.html Biographies/Hall.html] | ||
+ | |||
+ | Marshall Hall, "The Theory of Groups." Retrieved from [http://books.google.com/books?id=oyxnWF9ssI8C&lpg=PR1&pg=PA182#v=onepage&q&f=false id=oyxnWF9ssI8C&lpg=PR1&pg=PA182] | ||
− | + | Ronald Solomon. ''A Brief History of the Classification of the Finite Simple Groups'' [http://www.ams.org/journals/bull/2001-38-03/S0273-0979-01-00909-0/S0273-0979-01-00909-0.pdf www.ams.org/journals/bull/2001-38-03/S0273-0979-01-00909-0/S0273-0979-01-00909-0.pdf]<br> | |
− | + | Sikimeti Ma'u. ''Notes on Sylow's Theorems.'' Retrieved from [http://math.berkeley.edu/~sikimeti/SylowNotes.pdf math.berkeley.edu/~sikimeti/SylowNotes.pdf] | |
+ | Retrieved from [http://en.wikipedia.org/wiki/Sylow_theorems] | ||
+ | Retrieved from [http://en.wikipedia.org/wiki/Peter_Ludwig_Mejdell_Sylow] | ||
[[Category:MA453Fall2013Walther]] | [[Category:MA453Fall2013Walther]] |
Latest revision as of 18:00, 1 December 2013
Mark Rosinski, markrosi@purdue.edu Joseph Lam, lam5@purdue.edu Beichen Xiao, xiaob@purdue.edu
Contents
P-groups & Sylow Theorem
P-groups
Definitions:
- Let p be a prime p be an integer greater or equal to 0. A p-group is a group of order pn.
- A subgroup of order pk for some k ≥ 1 is called a p-subgroup.
- If |G| = pαm where p does not divide m, then a subgroup of order pα is called a Sylow p-subgroup of G.
Propositions:
If G is a p-group then G contains an element of order p.
- If G is a p-group then Z(G)cannot be equal to {1}
- Let p be a prime and let G be a group of order p2. Then G is abelian.
- If G is a p-group of order pa, then there exists a chain, {1} is contained in N1 contained in N2 contained in...contained in Na-1 contained in Gof normal subgroups of G, such that |Ni|=pi.
All content above and proofs of these Propositions can be found here
Further Information on p-groups:
- Lie Algebras
- A lie ring is a set R with two binary operations - addition and the Lie bracket - such that
- (R,+) is an abelian group;
- Bilinearity: The bracket operation distributes over addition;
- Alternating on the vector space g: [x,x] = 0 for all x in R;
- The Jacobi Identity: [[x,y],z]+[[y,z],x]+[[z,x],y]=0 for all x,y,z in R.
- If F is a field, and R is an F-vector space with a[x,y]=[ax,y] then R is a Lie algebra.
- To every finite p-group one can associate a Lie ring L(G), and if G/G' is abelian then L(G) is actually a lie algebra over the finite field GF(p).
- Proposition: Let φ be an automorphism of the finite p-group G. Then φ induces an automorphism on L(G), and if φ has order prime to p, then the induced automorphism has the same order.
- A lie ring is a set R with two binary operations - addition and the Lie bracket - such that
- Number of Groups
- Let g(n) denote the number of groups of order n.
- i) g(p)=1 for p a prime.
- ii) if p<q, then g(pq)=1 if q is not congruent to 1 mod p, and g(pq)=2 otherwise.
- iii) g(p2)=2.
- iv) g(p3)=5.
- Let g(n) denote the number of groups of order n.
From this we can see that the number of groups of order n depends more on the prime structure then on its size.
Look at this table to help explain this notion:
All content of Lie Algebras and Number of groups from this page can be found from here as well as additional information on these topics.
Automorphism Groups of order pn
Propositions:
- Let G denote the elementary abelian group of order pn. Then $ Aut(G)\cong GL_{n}(p) $, the group of n x n matrices over GF(p).
- Let G denote the cyclic group of order n. Then Aut(G) is abeilan, and has order φ(n), where φ denotes Euler's φ-fuction.
- Let G be a cyclic p-group with p odd, and write A=Aut(G). Then A is cyclic.
All content from this section is available from here as well as further information and proofs of these propositions on Automorphism Groups of order pn.
Classification of p-groups
1. Regular p-groups
Philip Hall, known for his work on group theory, initiated the Regular p-groups.
- Definitions:
- For every $ a, b \in G $ there exist $ c \in [<a,b>,<a,b>] $ such that apbp = (a'b)pcp
- For every $ a, b \in G $ there exist $ c_i \in [<a,b>,<a,b>] $ such that $ a^p b^p = (ab)^p c^p _1 c^p _2 . . . c^p _i $
- For evert $ a, b \in G $ and every positive integer n there exist elements $ c_i \in [<a,b>,<a,b>] $ such that $ a^q b^q = (ab)^q c^q _1 c^q _2 . . . c^q _i $ where q = pn
All content from this section can be found here and here
2. Powerful p-groups
- Definition:
- G is powerful if $ \mho _1 (G) \supseteq G' $
- Let $ N \triangleleft G $, then N is powerfully embedded in G if $ \mho _1 (N) \supseteq [N,G] $. G is powerful IFF it is powerfully embedded in itself. If N is powerfully embedded in G, then N is powerful, more importantly, if $ N \subseteq H \subseteq G $ and $ \frac{H}{N} $ is cyclic, then $ \mho _1 (H) \supseteq \mho _1 (N) \supseteq [N,H] = H' $, so H is powerful
- Proposition:
- G is powerful IFF G is powerfully embedded in G
- If N is powerfully embedded in G, then N is powerful and G centralizes $ \frac{N}{N^p} $
- If $ K \triangleleft G $ and N is powerfully embedded in G then $ \frac{N \times K}{K} $ is powerfully embedded in $ \frac{G}{K} $
- If N is powerfully embedded in G and $ x \in G $ then <x>N is powerful
- If M is powerfully embedded in G and N is powerfully embedded in G, then $ M \times N $ is powerfully embedded in G
All content from this section is available here and here as well as further information and proofs on Powerful p-groups
3. Special Groups
- Definition:
- Let G be a finite p-group, then G is defined to be special if either G is elementary abelian or G is of class 2 and G' = Φ(G) = Z(G) is elementary abelian. If G is a non-abelian special group with |Z(G)| = p, then G is defined to be extraspecial.
- Dihedral group $ D_{2n} = <x,y: x^{n} = y^{2} = 1, x^{y} = x^{\rightharpoondown1}> = <a,b: a^{2} = b^{2} = 1, (ab)^{n} = 1> $
- Quaternion group $ Q_{4n} = <x,y: x^{2n} = y^4 = 1, x^y = x^{\rightharpoondown1}, y^2 = x^n> $
- Semidihedral group $ SD_{2n} = <x,y: x^{2^{n-1}} = y^2 = 1, x^y = x^{2^{n-2}-1}> $
- Modular p-group $ Mod_n (p) = <x,y;: x^{p^{n-1}} = y^p = 1, x^y = x^{1+p^{n-2}} $
All content from this section is available here
Sylow's Theorems
Who is Sylow?
Peter Ludwig Mejdell Sylow (12 December 1832 – 7 September 1918) was a Norwegian mathematician who proved foundational results in group theory. He was born and died in Christiania (now Oslo).
Sylow was a high school teacher in Halden, Norway, from 1858 to 1898, and a substitute lecturer at Christiania University in 1862, covering Galois theory. It was then that he posed the question that led to his theorems regarding Sylow subgroups. Sylow published the Sylow theorems in 1872, and subsequently devoted eight years of his life, with Sophus Lie, to the project of editing the mathematical works of his countryman, Niels Henrik Abel.
He was appointed professor of Christiania University in 1898.
Notation:
Sylp(G) = the set of Sylow p-subgroups of G
np(G)= the # of Sylow p-subgroups of G =|Sylp(G)|
Theorems:
Let G be a group of order pαm, where p is a prime, m≥1, and p does not divide m. Then:
- Sylp(G) cannot be the empty set.
- All Sylow p-subgroups are conjugate in G. To expand, if P1 and P2 are both Sylow p-subgroups, then there is some g in G such that P1=gP1g-1. In particular, np(G)=(G:NG(P)).
- Any p-subgroup of G is contained in a Sylow p-subgroup
- np(G) is congruent to 1 mod p.
All content from this section and proofs of these Theorems can be found here
Application of Sylow theorems:
- In D6 all reflections are conjugate, as reflections correspond to Sylow 2-subgroups.
- In D12 reflections no longer correspond to Sylow 2-subgroups, and fall into two conjugacy classes.
Extra Information
For students looking for extensive history on p-groups, Sylow's Theorems and finite simple groups in general you can find this information here
References
Alavaro Loustau. Latex Tutorial. Retrieved from omega.albany.edu:8008/Symbols.html
Alexander Lubotzky. "Powerful p-Groups. I. Finite Groups." Retrieved from 0021869387902110/1-s2.0-0021869387902110-main.pdf
Bruno Kahn. "A Characterization of Powerfully Embedded Normal Subgroups of a p-Group." Retrieved from S0021869396968465/1-s2.0-S0021869396968465-main.pdf
Charles Richard Leedham-Green. "The Structure of Groups of Prime Power Order." Retrieved from id=34khoLiyP_QC&lpg=PA113&ots=4mE5X4ovFU&dq
David A. Craven. The Theory of p-Groups. Retrieved from craven/docs/lectures/pgroups.pdf
J J O'Connor. Philip Hall. Retrieved from Biographies/Hall.html
Marshall Hall, "The Theory of Groups." Retrieved from id=oyxnWF9ssI8C&lpg=PR1&pg=PA182
Ronald Solomon. A Brief History of the Classification of the Finite Simple Groups www.ams.org/journals/bull/2001-38-03/S0273-0979-01-00909-0/S0273-0979-01-00909-0.pdf
Sikimeti Ma'u. Notes on Sylow's Theorems. Retrieved from math.berkeley.edu/~sikimeti/SylowNotes.pdf
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