"Here's a thought" or "Final Showdown"
Okay, so I'm coming back to Ottawa on Monday the 22nd, and leaving on the 1st. Not counting the 1st ('cause I'm leaving then to go back for Gypsy), that makes 10 days. I also hand in my take-home exam on Friday, so that leaves Sat and Sun, two days, where I'm staying in Kingston to get some extra shifts in (read: $$$$). So that makes 12 days of my break. Total. I guess that makes it The Twelve Days Of Leski.
Now, for the mathie-babble - this STUPIDLY DIFFICULT take home exam for Group Theory/Abstract Algebra is driving met nuts - it's 8 questions long, 35% of my grade, and absurdly hard. Here's what I'm missing (recorded so one day I may actually know how to do these - and it's out of 115 marks):
3) Let G be a group of order 60: ["~=" mean "is isomophic to", group equivalence]
[5] a) Show that G ~= A5, or G has a normal Sylow 5-subgroup.
[5] b) Show that G ~= A5 or G ~= A4xZ5, or G has a cyclic normal subgroup of order 15.
[5] c) Show that PSL2(Z5) ~= A5 [I've got this one, we know that A5 is the only simple group of order 60, and we know that PSL2(Z5) is a simple group with 60 elements, so they must be isomorphic]
[5] d) How many groups of order 60 are there, up to isomorphism? [There's 13 of them, I know that - and I know what 10 of them could be, but I'm still missing 3]
[10] 4a) If p is an odd prime, show that there are five groups of order 2p^2 [I knew 4 of the groups right off - so I looked up online for the special case where |G|=18, and it said the 5th group is the semi-direct product of Z2 and (Z3xZ3). Funny thing is, we never learned how semi-direct products worked]
[5] 4b) If G is a group of order 2n with n being an odd number (greater than 1), show that G cannot be simple. [It's in the text, but the text's proof sucks]
[5] 4c) Let G be a simple group and |G| = 4pq where p and q are distinct primes. Show that G ~= A5 [so I have to show that p=3 and q=5, which isn't readily apparent].
So that's what I've got left to do. Something about Generators and Relations that was never fully explained in class comes in to most of these, but I simply don't see the connection. Worst part is I sucked out on the last 2 assignments, and I was hoping to do well in this course to balance out the rest of my marks. Damn.
Now, for the mathie-babble - this STUPIDLY DIFFICULT take home exam for Group Theory/Abstract Algebra is driving met nuts - it's 8 questions long, 35% of my grade, and absurdly hard. Here's what I'm missing (recorded so one day I may actually know how to do these - and it's out of 115 marks):
3) Let G be a group of order 60: ["~=" mean "is isomophic to", group equivalence]
[5] a) Show that G ~= A5, or G has a normal Sylow 5-subgroup.
[5] b) Show that G ~= A5 or G ~= A4xZ5, or G has a cyclic normal subgroup of order 15.
[5] c) Show that PSL2(Z5) ~= A5 [I've got this one, we know that A5 is the only simple group of order 60, and we know that PSL2(Z5) is a simple group with 60 elements, so they must be isomorphic]
[5] d) How many groups of order 60 are there, up to isomorphism? [There's 13 of them, I know that - and I know what 10 of them could be, but I'm still missing 3]
[10] 4a) If p is an odd prime, show that there are five groups of order 2p^2 [I knew 4 of the groups right off - so I looked up online for the special case where |G|=18, and it said the 5th group is the semi-direct product of Z2 and (Z3xZ3). Funny thing is, we never learned how semi-direct products worked]
[5] 4b) If G is a group of order 2n with n being an odd number (greater than 1), show that G cannot be simple. [It's in the text, but the text's proof sucks]
[5] 4c) Let G be a simple group and |G| = 4pq where p and q are distinct primes. Show that G ~= A5 [so I have to show that p=3 and q=5, which isn't readily apparent].
So that's what I've got left to do. Something about Generators and Relations that was never fully explained in class comes in to most of these, but I simply don't see the connection. Worst part is I sucked out on the last 2 assignments, and I was hoping to do well in this course to balance out the rest of my marks. Damn.