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Given the octic group $G = \{e, \sigma, \sigma^2, \sigma^3, \beta, \gamma, \delta, t\}$.

  1. Find a subgroup of $G$ that has order $2$ and is a normal subgroup of $G$.

  2. Find a subgroup of $G$ that has order $2$ and is not a normal subgroup of $G$.

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    you need some relations between b,g,d,t and s in order to do this. It is not clear from what you have written what happens when we compose b with itself. You may want to refer to http://en.wikipedia.org/wiki/Presentation_of_a_group to understand what information is necessary. Also when asking homework problems, its a good idea to indicate what you have tried and where you got stuck with the problem.2010-10-27
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    Like a table? I constructed a table for these.2010-10-27
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    When * beta by beta b*b = e the identity... should I show the cayley table I made?2010-10-27
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    @Jessica: You can specify it in the presentation of the group. Something like: G={ s,b,g,t | b^2 = e ... }, as the wiki article says.2010-10-27
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    You don't need to insert the whole Cayley table. If it is the case that b^2=e, g^2=e, d^2=e, t^2=e, you should edit that into the question. I'm not sure what you mean by your second comment. Adding the Cayley table would certainly suffice, but its not necessary if you specify the presentation2010-10-27
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    @Jessica Sorry for all the comments. If I understand your question correctly, the following hint may be helpful: consider the subgroup generated by s^2 and the subgroup generated by b.2010-10-27

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Each subgroup of order 2 is generated by an element of order 2, and vice versa. So listing all the elements of order 2 would be a good start.

It might also help you to show that a subgroup of order 2 is normal if and only if the element of order 2 that generates it is in the center of $G$.