How many subgroups does A4 have?
The group A4 has order 12, so its subgroups could have size 1, 2, 3, 4, 6, or 12. There are subgroups of orders 1, 2, 3, 4, and 12, but A4 has no subgroup of order 6 (equivalently, no subgroup of index 2). Here is one proof, using left cosets.
What is nontrivial subgroup?
A subgroup of a group is termed nontrivial, if the subgroup is not the trivial group, i.e. it has more than one element.
How many of the non trivial proper subgroups of S4 are normal?
Quick summary
| Item | Value |
|---|---|
| maximal subgroups | maximal subgroups have order 6 (S3 in S4), 8 (D8 in S4), and 12 (A4 in S4). |
| normal subgroups | There are four normal subgroups: the whole group, the trivial subgroup, A4 in S4, and normal V4 in S4. |
Which group has no proper normal subgroup?
simple group
A simple group is a group with no nontrivial proper normal subgroups.
How many subgroups of order 4 does A4 have?
one subgroup
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)〉.
Why A4 has no 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.
How do you find the non trivial proper subgroup?
A subgroup N of a group G is said to be proper if N≠G and to be non-trivial if N≠{e}, where e is the identity of G. For example N={0,2} is a proper subgroup of (Z/4Z,+), isomorphic to Z/2Z.
Which group has no non trivial subgroup?
Statement. If a nontrivial group has no proper nontrivial subgroup, then it is a cyclic group of prime order. In other words, it is generated by a single element whose order is a prime number.
Is A4 a normal subgroup of S4?
A4 is of Order 12, and therefore Index 2, hence A4 is Normal in S4.
How many subgroups are there in S4?
In all we see that there are 30 different subgroups of S4 divided into 11 conjugacy classes and 9 isomorphism types. As discussed, normal subgroups are unions of conjugacy classes of elements, so we could pick them out by staring at the list of conjugacy classes of elements.
Is A4 abelian?
Since S4/A4 is abelian, the derived subgroup of S4 is con- tained in A4. Also (12)(13)(12)(13) = (123), so that (nor- mality!) every 3-cycle is a commutator.
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.