## Subgroup and cosets of subgroups (1990)

Venue: | Journal of Formalized Mathematics |

Citations: | 46 - 9 self |

### BibTeX

@ARTICLE{Trybulec90subgroupand,

author = {Wojciech A. Trybulec},

title = {Subgroup and cosets of subgroups},

journal = {Journal of Formalized Mathematics},

year = {1990},

volume = {2},

pages = {2--2}

}

### Years of Citing Articles

### OpenURL

### Abstract

Summary. We introduce notion of subgroup, coset of a subgroup, sets of left and right cosets of a subgroup. We define multiplication of two subset of a group, subset of reverse elemens of a group, intersection of two subgroups. We define the notion of an index of a subgroup and prove Lagrange theorem which states that in a finite group the order of the group equals the order of a subgroup multiplied by the index of the subgroup. Some theorems that belong rather to [1] are proved.

### Citations

1278 | Tarski Grothendieck set theory
- Trybulec
- 1990
(Show Context)
Citation Context ...ls the order of a subgroup multiplied by the index of the subgroup. Some theorems that belong rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles =-=[10]-=-, [5], [15], [8], [12], [6], [13], [7], [3], [14], [1], [9], [16], [4], [17], [2], and [11] provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted str... |

1257 | Functions and their basic properties
- Byliński
- 1990
(Show Context)
Citation Context ...e theorems that belong rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], [5], [15], [8], [12], [6], [13], [7], [3], [14], [1], [9], [16], =-=[4]-=-, [17], [2], and [11] provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted structure, and A is a subset of G. Next we state the proposition (3) 1 If... |

1187 | Properties of subsets
- Trybulec
- 1990
(Show Context)
Citation Context ...r of a subgroup multiplied by the index of the subgroup. Some theorems that belong rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], [5], =-=[15]-=-, [8], [12], [6], [13], [7], [3], [14], [1], [9], [16], [4], [17], [2], and [11] provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted structure, and... |

988 | Relations and their basic properties
- Woronowicz
- 1990
(Show Context)
Citation Context ...p. Some theorems that belong rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], [5], [15], [8], [12], [6], [13], [7], [3], [14], [1], [9], =-=[16]-=-, [4], [17], [2], and [11] provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted structure, and A is a subset of G. Next we state the proposition (3)... |

725 | Some basic properties of sets
- Byliński
- 1990
(Show Context)
Citation Context ... order of a subgroup multiplied by the index of the subgroup. Some theorems that belong rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], =-=[5]-=-, [15], [8], [12], [6], [13], [7], [3], [14], [1], [9], [16], [4], [17], [2], and [11] provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted structur... |

642 | The fundamental properties of natural numbers
- Bancerek
- 1990
(Show Context)
Citation Context ...that belong rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], [5], [15], [8], [12], [6], [13], [7], [3], [14], [1], [9], [16], [4], [17], =-=[2]-=-, and [11] provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted structure, and A is a subset of G. Next we state the proposition (3) 1 If G is finit... |

634 | The ordinal numbers
- Bancerek
- 1990
(Show Context)
Citation Context ...group and prove Lagrange theorem which states that in a finite group the order of the group equals the order of a subgroup multiplied by the index of the subgroup. Some theorems that belong rather to =-=[1]-=- are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], [5], [15], [8], [12], [6], [13], [7], [3], [14], [1], [9], [16], [4], [17], [2], and [11] provide the... |

457 | Relations defined on sets
- Woronowicz
- 1990
(Show Context)
Citation Context ...orems that belong rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], [5], [15], [8], [12], [6], [13], [7], [3], [14], [1], [9], [16], [4], =-=[17]-=-, [2], and [11] provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted structure, and A is a subset of G. Next we state the proposition (3) 1 If G is ... |

311 | Finite sets
- Darmochwał
- 1990
(Show Context)
Citation Context ...ultiplied by the index of the subgroup. Some theorems that belong rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], [5], [15], [8], [12], =-=[6]-=-, [13], [7], [3], [14], [1], [9], [16], [4], [17], [2], and [11] provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted structure, and A is a subset o... |

303 | Families of sets
- Padlewska
- 1990
(Show Context)
Citation Context ... subgroup multiplied by the index of the subgroup. Some theorems that belong rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], [5], [15], =-=[8]-=-, [12], [6], [13], [7], [3], [14], [1], [9], [16], [4], [17], [2], and [11] provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted structure, and A is... |

292 | Domains and their Cartesian products
- Trybulec
- 1990
(Show Context)
Citation Context ...bgroup. Some theorems that belong rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], [5], [15], [8], [12], [6], [13], [7], [3], [14], [1], =-=[9]-=-, [16], [4], [17], [2], and [11] provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted structure, and A is a subset of G. Next we state the propositi... |

269 | Tuples, projections and Cartesian products
- Trybulec
- 1990
(Show Context)
Citation Context ...ng rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], [5], [15], [8], [12], [6], [13], [7], [3], [14], [1], [9], [16], [4], [17], [2], and =-=[11]-=- provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted structure, and A is a subset of G. Next we state the proposition (3) 1 If G is finite, then A ... |

251 | Vectors in real linear space
- Trybulec
- 1990
(Show Context)
Citation Context ...lied by the index of the subgroup. Some theorems that belong rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], [5], [15], [8], [12], [6], =-=[13]-=-, [7], [3], [14], [1], [9], [16], [4], [17], [2], and [11] provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted structure, and A is a subset of G. N... |

132 |
Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces
- Kusak
- 1990
(Show Context)
Citation Context ...y the index of the subgroup. Some theorems that belong rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], [5], [15], [8], [12], [6], [13], =-=[7]-=-, [3], [14], [1], [9], [16], [4], [17], [2], and [11] provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted structure, and A is a subset of G. Next w... |

39 | Zermelo theorem and axiom of choice
- Bancerek
- 1990
(Show Context)
Citation Context ... index of the subgroup. Some theorems that belong rather to [1] are proved. MML Identifier:GROUP_2. WWW:http://mizar.org/JFM/Vol2/group_2.html The articles [10], [5], [15], [8], [12], [6], [13], [7], =-=[3]-=-, [14], [1], [9], [16], [4], [17], [2], and [11] provide the notation and terminology for this paper. In this paper x is a set, G is a non empty 1-sorted structure, and A is a subset of G. Next we sta... |