Problems regarding group presentations and submodules

[Jösus]

Hi! I'm studying for an exam in group- and ring theory, and I have some questions about two problems that I have not managed to solve. I would greatly appreciate help.

Problem 1. Determine the order of the group G with the presentation $$(a,b \big\vert\: a^{6} = 1, b^{2} = a^{3}, ba = a^{-1}b)$$.

For the first problem, the relation $$ba = a^{-1}b$$ enables us to conclude that the order of $$G$$ is less than- or equal to 24 (we can collect all the a's to the left, and the largest possible orders for a and b are 6 and 4, respectively. Moreover, using the relation $$b^{2} = a^{3}$$ we can write all elements as $$a^{k}b^{j}$$ as $$k$$ lies in the range 0 to 5, and j is either 0 or 1. This shows that the group has order less than or equal to 12. Also, the group has order $$\geq 6$$, simply since we can construct a group of that order satisfying the relations (assume a is of order 3). My problem is that I cannot find clear arguments for why the group should have order 12, which I believe. Could I in some nice way just prove that for the collection of symbols $$a^{k}b^{j}$$ above, the group axioms hold? Does there perhaps exist a theorem concerning presentations of this type? Problem 2. Let M be the $$\mathbb{C}[x]$$-module $$\mathbb{C}^{3}$$ where $$\mathbb{C}$$ acts naturally and x acts via $$x \cdot a = T \cdot a$$, for elements $$a \in \mathbb{C}$$ and a given linear transformation T. How can one determine all submodules of M?For this problem less of an idea. I understand that the actions of $$\mathbb{C}$$ and x induces an action of any complec polynomial, and the submodules are clearily the subspaces of $$\mathbb{C}^{3}$$ stable under the linear map T. I guess some stable subspaces would be those spanned by eigenvectors for T. Are these all? I haven't thought a lot about this, but I'm running out of time, and would thus be very thankful for some help.

Thanks in advance!

 

[eok20]

Because the presentation is pretty nice, it's not too hard to show that you get a group by having elements of the form a^i b^j, i=0,...5, j=0,1 and specifying the multiplication rule. Alternatively, a semi-direct product of Z_3 and Z_4 has that presentation: http://shell.cas.usf.edu/~wclark/algctlg/small_groups.html