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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!
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!
