2. Tell me what you can about a group of order n Mike Krebs, Cal State LA This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

3. In this talk, I will describe a project I gave to an Abstract Algebra class. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

4. In this talk, I will describe a project I gave to an Abstract Algebra class. The project was simple. Each student was given a random three-digit number n. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

5. In this talk, I will describe a project I gave to an Abstract Algebra class. The project was simple. Each student was given a random three-digit number n. Each student was then instructed to “Tell me all you can about a group of order n.” This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

6. In this talk, I will describe a project I gave to an Abstract Algebra class. The project was simple. Each student was given a random three-digit number n. Each student was then instructed to “Tell me all you can about a group of order n.” More specifically: This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

7. Classify all abelian groups of order n. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

8. Classify all abelian groups of order n. Find an integer k such that a group G of order n must have a subgroup H of order k. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

9. Classify all abelian groups of order n. Find an integer k such that a group G of order n must have a subgroup H of order k. Is H always/sometimes/never normal in G? What are the possibilities for the number of conjugates that H can have? What else can you determine about the group H? If it’s normal, what can you say about the group G/H? How many such integers k can you find? This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

10. Classify all abelian groups of order n. Find an integer k such that a group G of order n must have a subgroup H of order k. Is H always/sometimes/never normal in G? What are the possibilities for the number of conjugates that H can have? What else can you determine about the group H? If it’s normal, what can you say about the group G/H? How many such integers k can you find? Is a group of order n always abelian? This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

11. Classify all abelian groups of order n. Find an integer k such that a group G of order n must have a subgroup H of order k. Is H always/sometimes/never normal in G? What are the possibilities for the number of conjugates that H can have? What else can you determine about the group H? If it’s normal, what can you say about the group G/H? How many such integers k can you find? Is a group of order n always abelian? Is a group of order n always simple? Sometimes simple? Never simple? This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

12. Is a group of order n always solvable? Sometimes solvable? Never solvable? This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

13. Is a group of order n always solvable? Sometimes solvable? Never solvable? Construct as many nonisomorphic nonabelian groups of order n as you can. Prove that they are not isomorphic. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

14. Is a group of order n always solvable? Sometimes solvable? Never solvable? Construct as many nonisomorphic nonabelian groups of order n as you can. Prove that they are not isomorphic. The Holy Grail: classify all groups of order of order n, up to isomorphism. (Depending on n, this might be trivial, or it might be extremely difficult.) This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

15. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA OK, let’s try it out in some gory detail. Let’s assign a random number. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

16. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA OK, let’s try it. Let’s assign a random number. We could use a random number generator, but let’s use some real technology. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

17. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

18. Classify all abelian groups of order n. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

19. Classify all abelian groups of order n. gap> Factors(n); This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

20. Classify all abelian groups of order n. gap> Factors(n); Example: 100 = 4 * 25 This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

21. Find an integer k such that a group G of order n must have a subgroup H of order k. What are the possibilities for the number of conjugates that H can have? Is H always/sometimes/never normal in G? If always, what can you say about the group G/H? How many such integers k can you find? This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

22. Find an integer k such that a group G of order n must have a subgroup H of order k. What are the possibilities for the number of conjugates that H can have? Is H always/sometimes/never normal in G? If always, what can you say about the group G/H? How many such integers k can you find? gap> l:=List([0..20],n->1+5*n); gap> List(l,x->RemInt(100,x)=0); This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

23. Example: c is congruent to 1 mod 5 c = 1, 6, 11, 16, 21, 26,... c divides 100. c = 1. So G has a unique normal Sylow 5-subgroup H. G/H has order 4, and H has order 25. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

24. Is a group of order n always simple? Sometimes simple? Never simple? (The answer will probably be never.) Is a group of order n always solvable? Sometimes solvable? Never solvable? (The answer will probably be always.) This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

25. Is a group of order n always simple? Sometimes simple? Never simple? (The answer will probably be never.) Is a group of order n always solvable? Sometimes solvable? Never solvable? (The answer will probably be always.) Indeed, that’s the case for n=100. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

26. Construct as many nonisomorphic nonabelian groups of order n as you can. (The semidirect product will be your friend here.) Prove that they are not isomorphic. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

27. Construct as many nonisomorphic nonabelian groups of order n as you can. (The semidirect product will be your friend here.) Prove that they are not isomorphic. Quick refresher: definition of semidirect product. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

28. Construct as many nonisomorphic nonabelian groups of order n as you can. (The semidirect product will be your friend here.) Prove that they are not isomorphic. Quick refresher: definition of semidirect product. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

29. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

30. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

31. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

32. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

33. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

34. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

35. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

36. You can tackle this question with pen-and-paper, or you can experiment using software. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

37. Let’s first construct a semidirect product using a = 2. (a = 1 yields the direct product of N and Q.) You can tackle this question with pen-and-paper, or you can experiment using software. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

38. gap> x:=(1,2,3,4,5); gap> A:=Group([x]); (cyclic group of order 5) gap> y:=(6,7,8,9,10); gap> B:=Group([y]); (cyclic group of order 5) gap> N:=DirectProduct(A,B); gap> z:=(1,2,3,4); gap> Q:=Group([z]); (cyclic group of order 4) Let’s first construct a semidirect product using a = 2. You can tackle this question with pen-and-paper, or you can experiment using software. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

39. gap> phi_2:=GroupHomorphismByImages(N,N, [x,y],[x^2,y^2]); gap> theta_2:=GroupHomorphismByImages(Q, AutomorphismGroup(N),[q],[phi_2]); gap> G_2:=SemidirectProduct(Q,theta_2,N); gap> IsAbelian(G_2); This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

40. gap> phi_2:=GroupHomorphismByImages(N,N, [x,y],[x^2,y^2]); gap> theta_2:=GroupHomorphismByImages(Q, AutomorphismGroup(N),[q],[phi_2]); gap> G_2:=SemidirectProduct(Q,theta_2,N); gap> IsAbelian(G_2); false This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

41. gap> phi_2:=GroupHomorphismByImages(N,N, [x,y],[x^2,y^2]); gap> theta_2:=GroupHomorphismByImages(Q, AutomorphismGroup(N),[q],[phi_2]); gap> G_2:=SemidirectProduct(Q,theta_2,N); gap> IsAbelian(G_2); false Similarly construct G_3, G_4 using a = 3, 4. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

42. gap> IsomorphismGroup(G_2,G_3); This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

43. gap> IsomorphismGroup(G_2,G_3); GroupHomomorphismByImages( Group( [ (2, 3,5,4)(6,11,21,16)(7,13,25,19)(8,15, 24,17)(9,12,23,20)(10,14,22,18), (1,6, 11,16,21)(2,7,12,17,22)(3,8,13,... This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

44. gap> IsomorphismGroup(G_2,G_3); GroupHomomorphismByImages( Group( [ (2, 3,5,4)(6,11,21,16)(7,13,25,19)(8,15, 24,17)(9,12,23,20)(10,14,22,18), (1,6, 11,16,21)(2,7,12,17,22)(3,8,13,... So G_2 and G_3 are isomorphic. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

45. gap> IsomorphismGroup(G_2,G_3); GroupHomomorphismByImages( Group( [ (2, 3,5,4)(6,11,21,16)(7,13,25,19)(8,15, 24,17)(9,12,23,20)(10,14,22,18), (1,6, 11,16,21)(2,7,12,17,22)(3,8,13,... So G_2 and G_3 are isomorphic. Now we have something to prove! (How could a human find an isomorphism from G_2 to G_3?) This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

46. gap> IsomorphismGroup(G_2,G_4); This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

47. gap> IsomorphismGroup(G_2,G_4); fail This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

48. gap> IsomorphismGroup(G_2,G_4); fail So G_2 and G_4 are not isomorphic. This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

49. gap> IsomorphismGroup(G_2,G_4); fail So G_2 and G_4 are not isomorphic. Now we have some thing else to prove! This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

50. “I remember a project that I did in one of my algebra classes where I came up with a way to classify the semi-direct products of certain groups by looking at conjugacy classes of matrices. Although I'd wager my ‘discovery’ is merely an instance of a more general established result, the feeling of being drawn to spend extra hours engaged with a problem, and being rewarded with finding heretofore unseen connections is exhilarating. The course ended and I had to move on to other things, but the experience affirmed for me the sheer pleasure of research.” This talk will (soon) be available by following the link for “Research and Talks” at: www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs