Question

Let \(G\) be a group with \(|G|=p^{2}\), where \(p\) is prime. Show that every proper subgroup of \(G\) is cyclic.

Short answer

Every proper subgroup of \( G \) is cyclic because they have order \( p \), where \( p \) is prime.

Step-by-step solution

Introduction to the Problem

We are given a group \( G \) with order \( |G| = p^2 \), where \( p \) is a prime. We need to demonstrate that every proper subgroup of \( G \) is cyclic. A cyclic group is one that can be generated by a single element.

Apply Sylow’s Theorems

Since \( |G| = p^2 \), we consider the applications of Sylow's theorems. By the first Sylow theorem, the number of \( p \)-Sylow subgroups, \( n_p \), divides \( |G| \) and is congruent to 1 mod \( p \). Thus \( n_p = 1 \) or \( p^2 \). If \( n_p = 1 \), there is only one \( p \)-Sylow subgroup which must be the entire group \( G \) itself.

Consider Subgroup Orders

Using Lagrange’s theorem, a subgroup of \( G \) must divide \( |G| \), so possible orders for subgroups are \( 1, p, p^2 \). The order of the whole group is \( p^2 \), so proper subgroups can have order 1 or \( p \).

Analyze Subgroups of Order p

Any subgroup of order \( p \) is a \( p \)-group, and since \( p \) is prime, such a subgroup must be cyclic. Each subgroup of order \( p \) is generated by any one of its non-identity elements due to the nature of being prime order.

Conclusion

Every proper subgroup of \( G \) must have an order of \( p \) or 1. Subgroups of order \( p \) are cyclic, as shown earlier. Therefore, every proper subgroup of \( G \) is cyclic.

Key concepts

Sylow's Theorems

Sylow's theorems are a collection of theorems in group theory that give us detailed information about the number and structure of subgroups within a group whose order is a power of a prime. They are especially useful when dealing with prime power orders, like the group mentioned in our exercise, where the order is \( |G| = p^2 \).

These theorems tell us how many such subgroups exist and provide conditions under which these subgroups are unique or normal. The key highlights include:

• The number of \( p \)-Sylow subgroups, \( n_p \), must divide \( |G| \) and is congruent to 1 modulo \( p \).
• If \( n_p = 1 \), then this subgroup is unique and necessarily normal.

In our case, since \( |G| = p^2 \), the first Sylow's theorem implies we examine conditions where \( n_p = 1 \) or \( p^2 \). If \( n_p = 1 \), the subgroup generated is unique, simplifying our work in classifying subgroups.

Lagrange's Theorem

Lagrange's theorem is a cornerstone in the study of finite group theory. It addresses the relationship between the order of a group and the order of its subgroups. If \( G \) is a finite group, the order of any subgroup \( H \) of \( G \) divides the order of the group. That is, \( |H| \mid |G| \).

In the context of our exercise, we have \( |G| = p^2 \), meaning the only possible orders for any subgroup are the divisors of \( p^2 \). These divisors are 1, \( p \), or \( p^2 \). Because we are interested in proper subgroups, we focus on 1 and \( p \).

This division essentially explains how subgroups are constrained in their possible sizes, which aligns with their possible structures.

Group Theory

Group theory is a mathematical framework for studying groups, which are abstract collections of elements with an operation satisfying certain axioms: closure, associativity, identity elements, and inverses. These principles allow for a cohesive and generalizable approach to understanding algebraic structures.

In our exercise, group theory provides the foundation for analyzing a group \( G \) of order \( p^2 \). Understanding groups and how they interact through subgroups and cosets is crucial.

When tackling problems in group theory, like identifying the cyclic nature of subgroups, we rely on abstract principles as outlined by the relevant theorems. Group theory allows us to organize these insights into a structured solution that can be applied to broader contexts.

Subgroups

Subgroups are subsets of a group that themselves satisfy the group axioms. They offer a way of breaking down the larger group into more manageable pieces. In our exercise, we're considering subgroups within a group \( G \) where \( |G| = p^2 \).

Possible subgroups here can have orders that are divisors of \( p^2 \), and proper subgroups have orders of 1 or \( p \).

Especially for groups of prime power order, like those of order \( p \), subgroups are simple: any subgroup of prime order is necessarily cyclic. This insight directly comes from the nature of groups with prime order and illustrates why every proper subgroup is cyclic.