- #1
Pratibha
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How to find the composition series of a group of order 30
Composition Series for Group Order 30
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- Thread starter Pratibha
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- Composition Series
In summary, to find the composition series of a group of order 30, you can use the Abelian's groups structure theorem if the group is abelian, or you can follow the steps of finding a normal subgroup of index 2 and using the Sylow theorems to construct a composition series with factors of orders 2, 3, and 5. Additionally, for finitely generated abelian groups, the Jordan-Holder theorem states that all composition series have the same factors, which can be ordered in a specific way.
- #2
Fallen Angel
- 202
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Hi,
It may depend on the caharacteristics of the group.
If your group it's abelian then use the Abelian's groups structure theorem.
If not, I think we need to know something more.
It may depend on the caharacteristics of the group.
If your group it's abelian then use the Abelian's groups structure theorem.
If not, I think we need to know something more.
- #3
Pratibha
- 8
- 0
what is statement of abelian structure theorem?
- #4
johng1
- 235
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This may be beyond your knowledge, but here goes:
A group of order 30 has a normal subgroup of index 2 (Any group of even order with a cyclic Sylow 2 subgroup has a normal subgroup of index 2; proved by considering the Cayley representation of G as a permutation group.) So let H be normal of order 15. By the Sylow theorems H has a normal Sylow 5 subgroup K (cyclic) and a normal Sylow 3 subgoup L. So one composition series (actually a chief series--all subgroups in series are normal in G) is:
$$<1>\,\trianglelefteq L\trianglelefteq H\trianglelefteq G$$ with the factors cyclic of orders 2, 3 and 5. The Jordan-Holder theorem says all composition series have the same factors.
A group of order 30 has a normal subgroup of index 2 (Any group of even order with a cyclic Sylow 2 subgroup has a normal subgroup of index 2; proved by considering the Cayley representation of G as a permutation group.) So let H be normal of order 15. By the Sylow theorems H has a normal Sylow 5 subgroup K (cyclic) and a normal Sylow 3 subgoup L. So one composition series (actually a chief series--all subgroups in series are normal in G) is:
$$<1>\,\trianglelefteq L\trianglelefteq H\trianglelefteq G$$ with the factors cyclic of orders 2, 3 and 5. The Jordan-Holder theorem says all composition series have the same factors.
- #5
Fallen Angel
- 202
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Every finitely generated abelian group $G$ is isomorphic to a direct product
$\Bbb{Z}^{n}\oplus \Bbb{Z}_{a_{1}}\oplus \ldots \oplus \Bbb{Z}_{a_{k}}$
Moreover, we can order it in such a way that $a_{i}$ divides $a_{i+1}$ for every $i=1,\ldots, k-1$
PS: It's easier than I thought using the way johng has just posted
$\Bbb{Z}^{n}\oplus \Bbb{Z}_{a_{1}}\oplus \ldots \oplus \Bbb{Z}_{a_{k}}$
Moreover, we can order it in such a way that $a_{i}$ divides $a_{i+1}$ for every $i=1,\ldots, k-1$
PS: It's easier than I thought using the way johng has just posted
FAQ: Composition Series for Group Order 30
What is a Composition Series for Group Order 30?
A Composition Series for Group Order 30 is a way to break down a group with 30 elements into a series of smaller groups, each of which is a normal subgroup of the previous group. This series provides a way to understand the structure of the group and its subgroups.
How many groups are in a Composition Series for Group Order 30?
A Composition Series for Group Order 30 will have at least 3 groups, and at most 5 groups. This is because 30 is a highly composite number, meaning it has many divisors, and each divisor will give a different number of groups in the series.
What is the significance of a Composition Series for Group Order 30?
A Composition Series for Group Order 30 is significant because it provides information about the structure of the group. It can help determine if the group is simple (has no nontrivial normal subgroups) or if it can be broken down into smaller groups. This information is useful in many areas of mathematics and physics.
How can one determine a Composition Series for Group Order 30?
To determine a Composition Series for Group Order 30, one can use a process called the Jordan-Hölder Theorem. This involves finding normal subgroups of the group and then breaking it down into smaller groups until reaching the trivial subgroup. The resulting series will be a Composition Series for Group Order 30.
Can a Composition Series for Group Order 30 be applied to other group orders?
Yes, a Composition Series can be applied to any group with a finite order. However, the number of groups in the series will vary depending on the order of the group. For example, a Composition Series for a group with order 60 will have at least 3 groups, and at most 7 groups.