 Group theory

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# Conclusion

Following completion of this free OpenLearn course, Group theory, as well as extending your understanding of group theory, you should also find that your skills and confidence in reading, understanding and writing mathematical arguments are improving.

You should now be able to:

• apply the Internal Direct Product Theorem in simple cases
• decide whether a given group is cyclic, and given a finite cyclic group, find a generator for a subgroup of a given order
• express a given finite cyclic group as the direct product of cyclic groups of prime power order and, given two direct products of cyclic groups, determine whether or not they are isomorphic
• express products of elements of a group defined by generators and relations in appropriate standard form
• recognise the dihedral and dicyclic groups when described using a standard form.

This free OpenLearn course is an extract from the Open University course M303 Further pure mathematics [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] .

If you feel you are ready to move on in your study of pure mathematics but don’t have time to study a full Open University course at this time, you might like to study one of the free OpenLearn courses Number theory, Rings and polynomials and Metric spaces and continuity. They are further adapted extracts from the Open University course M303 Further pure mathematics.

If you are interested in applied mathematics, you might like to study the free OpenLearn courses Introduction to the calculus of variations, Linear programming – the basic ideas and Kinematics of fluids.

If you are interested in statistics, you might like to study the free OpenLearn courses Modelling events in time, Univariate continuous distribution theory and Point estimation.

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