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Math Help - Finding the cyclic subgroup...

  1. #1
    Member elninio's Avatar
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    Finding the cyclic subgroup...

    Find the cyclic subgroup generated by
    2 1
    0 2
    (matrix)
    in GL_2(Z_3).

    I suppose im confused as to what exactly it's asking me to do.
    GL_2(Z_3) signifies 2x2 matrices with mod3 entried, correct? How do I find the cyclic subgroup?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by elninio View Post
    Find the cyclic subgroup generated by
    2 1
    0 2
    (matrix)
    in GL_2(Z_3).

    I suppose im confused as to what exactly it's asking me to do.
    GL_2(Z_3) signifies 2x2 matrices with mod3 entried, correct? How do I find the cyclic subgroup?
    The cyclic subgroup generated by the element a is all elements of the form a^n for n \in \mathbb{N}. So, just find A, A^2, A^3, etc until you get back to the identity. The powers of your generator will be your group.
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  3. #3
    Member elninio's Avatar
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    Thank you! I've found the answer. Its too long to write out but it contains six matrices.
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  4. #4
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    Quote Originally Posted by elninio View Post
    Thank you! I've found the answer. Its too long to write out but it contains six matrices.

    Correct.

    Tonio
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