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Whilst answering my homework, I have noticed that a question has said that I need to consider the subgroups of $A=\mathbb{Z}_{135}$, including $W=\langle 18,81\rangle$. I was under the impression that for a subset to be a subgroup $a+b$ in the subgroup had to be in the subgroup as well. $18+81 = 99\mod{135}$ which isn't in the subgroup.

Is it my reasoning or the problem sheet that is wrong? Thank you in advance.

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The notation $<H>$ usually refers to the subgroup generated by the set $H \subseteq G$ where $G$ is a group. –  Adrián Barquero Dec 13 '10 at 2:20
    
Thank you! The question makes far more sense now. I'm glad it's just notation and not all my beliefs about group theory being taken away! –  Lucy Marshall Dec 13 '10 at 2:26
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The question got answered in a comment and the answer got commented in an answer... Where is the world coming to! –  Mariano Suárez-Alvarez Feb 11 '11 at 4:21
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For the sake of having an answer, Adrián's comment:

The notation $\langle H \rangle$ usually refers to the subgroup generated by the set $H \subseteq G$ where $G$ is a group.

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