Homework

Hello D456. Those don't look like questions. Did you receive any instructions or additional information?

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In the first question: \(\mathcal{G}\) is a group and \(a\in\mathcal{G}\). The usual definition is \(\{g: ga=ag,~g\in\mathcal{G}\}\) .
Clearly \(c(a)\ne\emptyset\) because the identity is in \(c(a)\). Is it true that \(a\in c(a)~?\)
If you know that \(ga=ag\) does it follow that \(gag^{-1}=a~?\)
 
Darling456 said:
I need help with my homework. I don't know what to do.
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Subhotosh Khan said:
Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING

Please share your work/thoughts about this problem.
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I choose to answer this post in two parts because it involves two totally different ideas (areas of mathematics).
The set of \(\mathcal{M}_{2\times 2}:~2\times 2\) matrices with real/complex entries with addition as the operation form a group.
In this posting, \(H= \left\{ {\left[ {\begin{array}{*{20}{c}} a&0 \\ 0&b \end{array}} \right]:a + b = 0} \right\}\) is to be shown as a subgroup of \(\mathcal{M}_{2\times 2}\)
This is from Kenneth Miller's text:Theorem: if H is a subset of a group G then H is a subgroup of G if and only if
for each \(\left\{ {a,b} \right\} \subseteq H\) then \(a{b^{ - 1}} \in H\)
 
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