Prove that a group which does not have proper subgroups is either an identity group (e) or is of prime order, where e is the identity element of the group.

### Proof:

Let \( G \) be a group that does not have proper subgroups. We want to prove that \( G \) is either the identity group (a group consisting solely of the identity element) or a group of prime order.

#### Case 1: \( G \) is the identity group
If \( G = \{e\} \), where \( e \) is the identity element, then \( G \) has only one element. In this case, \( G \) is trivially a group, and since it only contains the identity element, it has no proper nontrivial subgroups. Thus, this case satisfies the condition that \( G \) has no proper subgroups.

#### Case 2: \( G \) has more than one element
Now, assume that \( G \) contains more than one element. We claim that in this case, the order of \( G \), denoted by \( |G| \), must be a prime number.

Let \( |G| = n \). By Lagrange's theorem, the order of any subgroup \( H \) of \( G \) must divide \( n \). This means the order of any possible subgroup of \( G \) must be a divisor of \( n \).

Since we are given that \( G \) does not have any proper subgroups, the only subgroups of \( G \) are \( G \) itself and the trivial subgroup \( \{e\} \). This implies that no divisor of \( n \) other than \( 1 \) (corresponding to \( \{e\} \)) and \( n \) (corresponding to \( G \)) can exist. Therefore, \( n \) must have no divisors other than 1 and itself, which is the definition of a prime number.

Hence, \( |G| \) must be prime.

### Conclusion:
Thus, we have proven that if a group \( G \) does not have proper subgroups, then \( G \) is either the identity group (when \( |G| = 1 \)) or a group of prime order (when \( |G| = p \), where \( p \) is a prime number).

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Here are 5 related questions for further exploration:
1. What is Lagrange's theorem in group theory?
2. Can a group of composite order have no proper subgroups?
3. What is an example of a group of prime order?
4. How does the concept of normal subgroups relate to this proof?
5. What is the significance of prime numbers in group theory?

**Tip:** Understanding the divisors of the group's order is key in many group-theoretic results, especially when using Lagrange’s theorem.

Explore a detailed proof that a group with no proper subgroups is either an identity group or has prime order. This solution leverages Lagrange's theorem and discusses prime numbers in group theory. Suitable for undergraduate math students.

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