What is the significance of a group with no proper subgroups?

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    2015
In summary, a group with no proper subgroups is a mathematical structure with a set of elements and a binary operation that satisfies certain properties. This type of group is significant because it is a fundamental concept with various applications and allows for efficient computations and proofs. It differs from other groups by being the most basic type and having no non-trivial substructures. It cannot have any subgroups and can be represented or visualized using a Cayley graph or a multiplication table.
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Ackbach
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Here is this week's POTW:

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Suppose $G$ is a group with no proper subgroups. What can be said about $G?$ Prove your statements.

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Congratulations to johng for his correct solution to this week's POTW, which follows:

$G$ is finite and cyclic of order 1 or cyclic of order a prime.
Let $E=\langle 1\rangle$ be the identity subgroup of $G$ and suppose $G\neq E$. Let $x\in G$ with $x\neq 1$. Then $\langle x\rangle=G$ and $x$ has finite order, for otherwise $\langle x^2\rangle$ is a proper subgroup of $G$. Suppose $x$ has order $n$ with $n$ composite, say $n=mq$. Then $\langle x^m\rangle$ has order $q$ and so would be a proper subgroup of $G$. Hence $n$ is prime. Thus $G=\langle x\rangle$ is cyclic of prime order $n$.
 

FAQ: What is the significance of a group with no proper subgroups?

What exactly is a group with no proper subgroups?

A group with no proper subgroups is a mathematical structure consisting of a set of elements and a binary operation that satisfies certain properties, such as closure, associativity, identity, and inverse. This means that every element in the group can be combined with any other element using the operation to produce another element in the group.

Why is this type of group significant?

A group with no proper subgroups is significant because it is a simple and fundamental mathematical concept that has applications in various fields, such as abstract algebra, number theory, and cryptography. It also allows for more efficient computations and proofs in mathematics.

How does a group with no proper subgroups differ from other groups?

A group with no proper subgroups differs from other groups in that it is the most basic type of group and has no non-trivial substructures. This means that all of its elements are connected and cannot be broken down into smaller groups.

Can a group with no proper subgroups have any subgroups at all?

No, a group with no proper subgroups cannot have any subgroups. This is because a subgroup must contain at least two elements and have the same binary operation as the original group, which is not possible if the group has no proper subgroups.

How can a group with no proper subgroups be represented or visualized?

A group with no proper subgroups can be represented or visualized using a Cayley graph, which is a mathematical structure that shows the elements of a group and the relationships between them. It can also be represented using a multiplication table, which lists all possible combinations of elements and their resulting products.

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