## 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

Overview | 1 |

What do groups look like? | 11 |

Why study groups? | 25 |

Algebra at last | 41 |

Five families | 63 |

Subgroups | 97 |

Products and quotients | 117 |

The power of homomorphisms | 157 |

Sylow theory | 193 |

Galois theory | 221 |

A Answers to selected Exercises | 261 |

285 | |

### Common terms and phrases

abelian groups algebraic answer arithmetic automorphism blue arrows C2 x C2 C3 x C3 called Cauchy's Theorem Cayley diagram Chapter clockwise codomain column compute configuration conjugacy classes conjugate connect contains corresponding cosets of H create cycle graph cyclic groups Definition 3.1 describe diagram in Figure dihedral groups direct product group domain example Exercise factor Galois group group C2 group G group S3 group theory groups of order Hasse diagram Hint homomorphism horizontal flip identity element illustrated inverse isomorphic left cosets mathematical move multiplication table Ng(H nodes non-abelian groups normal subgroups orbit pattern permutations polynomial equations proof quotient map quotient process rational numbers real numbers rearrange rectangle puzzle result right cosets roots rotation Rubik's Cube Section semidirect product shown in Figure shows solvable solve Stab(s stable elements step structure subgroup H subgroup of order Sylow p-subgroup Sylow Theorem symmetry technique vertical flip visual whole numbers