## How many subgroups of order 4 does A4 have?

Table of Contents

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Table classifying subgroups up to automorphism

Automorphism class of subgroups | List of subgroups | Total number of subgroups |
---|---|---|

V4 in A4 | 1 | |

A3 in A4 | , , , | 4 |

whole group | all elements | 1 |

Total (5 rows) | — | 10 |

## Does A4 have a subgroup of order 4?

In A4 there is one subgroup of order 4, so the only 2-Sylow subgroup is {(1), (12)(34), (13)(24), (14)(23)} = 〈(12)(34),(14)(23)〉. There are four 3-Sylow subgroups: {(1), (123), (132)} = 〈(123)〉, {(1), (124), (142)} = 〈(124)〉, {(1), (134), (143)} = 〈(134)〉, {(1), (234), (243)} = 〈(234)〉.

**How many elements are there in A4?**

Summary

Item | Value |
---|---|

order of the whole group (total number of elements) | 12 (see order computation for more) |

conjugacy class sizes | 1,3,4,4 maximum: 4, number: 4, sum (equals order of whole group): 12, lcm: 12 See conjugacy class structure for more. |

number of conjugacy classes | 4 See number of conjugacy classes for more. |

**How many subgroups does S4 have?**

In all we see that there are 30 different subgroups of S4 divided into 11 conjugacy classes and 9 isomorphism types.

### Does A4 have a subgroup of order 6?

But A4 contains 8 elements of order 3 (there are 8 different 3-cycles), and so not all elements of odd order can lie in the subgroup of order 6. Therefore, A4 has no subgroup of order 6.

### Does A4 have an element of order 4?

But A4 doesn’t contain any elements of order 4, so H must contain only elements of order 1, 2. The only elements of order 2 are (12)(34), (13)(24), and (14)(23) and we see that 11, (12)(34), (13)(24), (14)(23)l is indeed a subgroup (of order 4).

**What are the 12 elements of A4?**

Elements of A4 are: (1), (1, 2,3), (1,3, 2), (1, 2,4), (1,4,2), (1,3,4), (1,4,3), (2,3,4), (2,4,3), (1, 2)(3,4), (1,3)(2,4), (1,4)(2,3). (Just checking: the order of a subgroup must divide the order of the group. We have listed 12 elements, |S4| = 24, and 12 | 24.) Show that φ is a homomorphism.

**Why A4 has no subgroup of order 6?**

## What are the subgroups of Z4?

Table classifying subgroups up to automorphism

Automorphism class of subgroups | List of subgroups (power notation, generator ) | Order of subgroups |
---|---|---|

trivial subgroup | 1 | |

Z2 in Z4 | 2 | |

whole group | 4 | |

Total (3 rows) | — | — |

## What are the possible orders of elements in A4?

So, the possible orders of elements in A4 are 1, 2, 3. (c) The possible cycle types of elements in S5 are: identity, 2-cycle, 3-cycle, 4-cycle, 5-cycle, product of two 2-cycles, a product of a 2-cycle with a 3- cycle.

**How many 3 cycles does A4 have?**

8 different 3-cycles

But A4 contains 8 elements of order 3 (there are 8 different 3-cycles), and so not all elements of odd order can lie in the subgroup of order 6.