## Visual Group TheoryThis text approaches the learning of group theory visually. It allows the student to see groups, experiment with groups and understand their significance. It brings groups, subgroups, homomorphisms, products, and quotients into clear view. Every topic and theorem is accompanied with a visual demonstration of its meaning and import, from the basics of groups and subgroups through advanced structural concepts such as semidirect products and Sylow theory. Opening chapters anchor the reader's intuitions with puzzles and symmetrical objects, defining groups as collections of actions. This approach gives early access to Cayley diagrams, the visualization technique central to the book, due to its unique ability to make group structure visually evident. This book is ideal as a supplement for a first course in group theory or alternatively as recreational reading. |

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### Contents

Acknowledgments | 1 |

4 | 35 |

6 | 94 |

8 | 148 |

Sylow theory | 193 |

Galois theory | 221 |

A Answers to selected Exercises | 261 |

285 | |

### Common terms and phrases

abelian groups action algebraic answer appears apply arrows begin called Cayley diagram Chapter column combining common complex compute configuration conjugate connect Consider contains copy corresponding create cube cycle cyclic groups Definition describe direct product divide domain elements equation example Exercise explain Explorer expression extension fact factor field Figure flip four Galois give given group theory groups of order homomorphism horizontal identity illustrated isomorphic lead left cosets look mathematical means move multiplication table nodes normal subgroups numbers objects operation orbit original path pattern permutations polynomial possible prime proof prove puzzle question quotient reason rectangle represent requires result rewiring roots rotation Rule satisfies semidirect product shows solve step structure subgroup Sylow symmetry technique tell Theorem true turn vertical visual write