Abstract Algebra IBL Wiki

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====== Some explicit examples of abstract groups ======
==== Problem ====
Give examples of [[definition:Group|groups]] with the following properties by $\textbf{explicitly}$ giving a set and defining both functions as well as the distinguished element. 
  - A group with $3$ elements
  - A different group with $3$ elements, if possible
  - A group with $4$ elements
  - A different group with $4$ elements, if possible
  - An infinite group

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==== Remarks ====  
  * None.

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==== $\LaTeX$ version ====
<file tex problem.explicit_examples_of_abstract_groups.tex>
%%%%%
% DEPENDENCIES 
%%%%%
\begin{problem}
Give examples of groups with the following properties by \textbf{explicitly} giving a set and defining both functions as well as the distinguished element. 
\begin{enumerate}
\item A group with $3$ elements
\item A different group with $3$ elements, if possible
\item A group with $4$ elements
\item A different group with $4$ elements, if possible
\item An infinite group
\end{enumerate}
\end{problem}
</file>

==== Questions ====
Did you decide to use your definition that used to be in the wiki? I would potentially change this to be 
  *by $\textbf{explicitly}$ giving a set, defining the binary operation, stating the inverse of each element, and identifying the identity.
  *Do you want them to show that the group axioms hold? I could see a student doing all the above but failing to check associativity.  They'll get lucky most likely with small sets...

{{tag>problem needsreview rben}}

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