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    Hi everyone, I am new to the forum.

    I have the following Cayley Table

    The table is for an equilateral triangular prism, with three identical rectangular faces.

    I have found the twelve symmetries and I am now asked to find subgroups of order 2, 3 and 4.

    I have found the subgroups of order 2 and 3 straight away (by just trying different elements together) but I am struggling to find a subgroup that has order 4.

    Is there a way to find it as I am having trouble producing a full Cayley table?


    I have listed the symmetry group as {e,a,b,c,d,f,r,s,t,u,v,w}

    e is the identity
    a and b are rotations in the vertical axis
    c, d and f are rotations about the horizontal axis (going through the rectangles)
    r,s,t are the reflections cutting through the vertical plane
    u is the reflection in the horizontal plane that cuts through the centre
    v and w are a composition of u and a or b.

    I hope I've explained that all correctly. Let me know if you need any more info.


    **********The attached Cayley table I found on an old post of this forum. It is for an equilateral triangular prism with 3 square faces. I've added that incase that can help in some way******

    The elements are:

    r1 and r2 are rotations.

    r3, r4 and r5 are the other rotations.

    p1, p2, p3 are reflections.

    p4 is the other reflection.

    c1 is p4 followed by r1.

    c2 is p4 followed by r2
    Attached Images
     
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    (Original post by makin)

    I have listed the symmetry group as {e,a,b,c,d,f,r,s,t,u,v,w}

    e is the identity
    a and b are rotations in the vertical axis
    c, d and f are rotations about the horizontal axis (going through the rectangles)
    r,s,t are the reflections cutting through the vertical plane
    u is the reflection in the horizontal plane that cuts through the centre
    v and w are a composition of u and a or b.

    I hope I've explained that all correctly. Let me know if you need any more info.
    Welcome to :tsr:

    I was a bit confused, until I realised the end faces of your prism were vertical, rather than horizontal.

    So, up to isomorphism there are only two groups of order 4.

    The cyclic group \mathrm{Z}_4 requiring an element of order 4.

    And the Klein four-group \mathrm{Z}^2_2, also written \mathrm{Z}_2\times\mathrm{Z}_2

    So, your subgroup must be isomorphic to one of these two.

    Does that give you any ideas to investigate?
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    Or look at it this way: you (should) know that the order r of an element of a subgroup of order 4 divides 4. Therefore your subgroup is either generated by one element of order 4, or contains the identity plus three elements of order two.

    Thus you either need to find an element of order four or pick out three non-identity order two elements and check that they form a group. This narrows down the search enough to do it by hand.
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    Thanks to both of you. Those comments have helped me get my head around it
 
 
 
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