Intereting Posts

Why rationalize the denominator?
Tarski Monster group with prime $3$ or $5$
Elements of order 5 in $S_7$, odd permutations of order 4 in $S_4$, and find a specific permutation in $S_7$
$1 + 1 + 1 +\cdots = -\frac{1}{2}$
Orthogonal trajectories for a given family of curves
Prove that $\frac{7}{12}<\ln 2<\frac{5}{6}$ using real analysis
Are monomorphisms of rings injective?
Algebra: What allows us to do the same thing to both sides of an equation?
Is there a concept of |x|<0?
Geometric basis for the real numbers
Domain is Hausdorff if image of covering map is Hausdorff
Relative error of machine summation
Positive series problem: $\sum\limits_{n\geq1}a_n=+\infty$ implies $\sum_{n\geq1}\frac{a_n}{1+a_n}=+\infty$
How are we able to calculate specific numbers in the Fibonacci Sequence?
Proving that every vector space has a norm.

If $G$ is a nonabelian group of order $p^3$ for $p$ a prime, and every nonidentity element has order $p$, does there exist a subgroup isomorphic to $\mathbb{Z}_p\times\mathbb{Z}_p$?

Based on some searching, I bet it’s true. I read a construction of such a subgroup by taking a element $x$ of order $p\in Z(G)$. Then you can “pull back to $G$ a subgroup of $G/\langle x\rangle$ of order $p$. The resulting subgroup will be normal with index $p$, and isomorphic to $\mathbb{Z}_p\times\mathbb{Z}_p$ since it’s an abelian group killed by $p$.”

This writing is all very vague to me. Is there a clearer explicit explanation of why a subgroup of order $p^2$ isomorphic to $\mathbb{Z}_p\times\mathbb{Z}_p$ exists in $G$?

- Extension of a group homomorphism
- Calculating the norm of an element in a field extension.
- The preimage of a maximal ideal is maximal
- If $R$ is a local ring, is $R]$ (the ring of formal power series) also a local ring?
- Prove that the following are isomorphic as groups but not as rings
- Toric Varieties: gluing of affine varieties (blow-up example)

- How to prove that homomorphism from field to ring is injective or zero?
- Finitely generated ideal in Boolean ring; how do we motivate the generator?
- Infinite Degree Algebraic Field Extensions
- Proving a ring is Noetherian when all maximal ideals are principal generated by idempotents
- Finding the kernel of ring homomorphisms from rings of multivariate polynomials
- Union of the conjugates of a proper subgroup
- Is this an equivalent statement to the Fundamental Theorem of Algebra?
- Ring homomorphism defined on a field
- Splitting field of $x^{n}-1$ over $\mathbb{Q}$
- Video lectures on Group Theory

**Theorem.** Let $p$ be a prime, and let $G$ be a group of order $p^n$. Then $G$ has subgroups of order $p^i$ for each $i$, $0\leq i\leq n$.

**Lemma.** Let $G$ be a group of order $p^n$, where $p$ is a prime. Then $Z(G)\neq\{1\}$.

*Proof of Lemma.* This follows form the class formula: define an equivalence relation on $G$ letting $a\sim b$ if and only if there exists $g\in G$ such that $gag^{-1}=b$. Since $gag^{-1}=hah^{-1}$ if and only if $h^{-1}ga = ah^{-1}g$, if and only if $h^{-1}g\in C_G(a)$, if and only if $h$ and $g$ represent the same coset modulo $C_G(a)$, it follows that the equivalence class of $a$ has as many elements as the index of $C_G(a)$.

Now let $a_1,\ldots,a_k$ be representatives from the equivalence classes of $G$. Then

$$p^n = |G| = \sum_{i=1}^k|\{b\in G\mid a_i\sim b\}| = \sum_{i=1}^k[G:C_G(a_i)].$$

Now, there is at least one element whose centralizer is all of $G$: the identity; and since only the identity is related to $e$, one of the $a_i$ must equal $e$. Thus, the sum on the right contains terms that are not congruent to $0$ modulo $p$. But the sum is $0$ modulo $p$, so there is more than one $i$ for which $[G:C_G(a_i)] = 1$. In particular, there is an $a_i\neq e$ such that $C_G(a_i)=G$, which means $a_i\in Z(G)$. $\Box$

*Proof of Theorem.* Induction on $n$. If $|G|=p$, then $G$ is cyclic of order $p$, and has subgroups of orders $1$ and $p$.

Assume by induction the result holds for groups of order $p^n$, and let $G$ be a group of order $p^{n+1}$. Let $a\in Z(G)$ be an element of order $p$ (it exists, since $Z(G)$ is not trivial), and let $N=\langle a\rangle$. Then $N\triangleleft G$, since for all $g\in G$, $gag^{-1}=a$; so $K=G/N$ is a group of order $p^n$. By the induction hypothesis, $K$ has subgroups $K_0,\ldots,K_n$ of order $p^0,p^1,\ldots,p^n$, respectively. For each $i$, let $H_i = \{g\in G\mid gN\in K_i\}$. By the isomorphism theorems, $H_i$ are subgroups of $G$, and $H_i/N\cong K_i$; therefore, $|H_i|=|K_i||N| = p^i\times p = p^{i+1}$, so $G$ has subgroups $H_0,\ldots,H_n$ of orders $p^1,\ldots,p^{n+1}$. Together with the identity, we get that $G$ has subgroups of order $p^i$ for $i=0,\ldots,n+1$, as desired. $\Box$

**Corollary.** Let $G$ be a group of order $p^n$, $p$ a prime, and $n\gt 1$, in which every element is exponent $p$. Then $G$ contains a subgroup isomorphic to $C_p\times C_p$.

*Proof.* Let $H$ be a subgroup of $G$ of order $p^2$ (which exists by the Theorem). Then $H$ is of order $p^2$, so the center has order at least $p$; but $H/Z(H)$ cannot be cyclic and nontrivial, so $Z(H)$ cannot be of order exactly $p$, hence $Z(H)=H$ and $H$ is abelian. Thus, $H$ is an abelian group of order $p^2$ in which every element is order $p$; the only possibility is $H\cong C_p\times C_p$. $\Box$

The Sylow theorems may be overkill here, but they guarantee that a group of order $p^3$ has a subgroup of order $p^2$. Now up to isomorphism there are only 2 groups of order $p^2$, the cyclic group and the product of two cyclic groups. Since you hypothesize every non-identity element has order $p$, it can’t be the cyclic group, so it has to be the other one, and we’re done.

Let $$

x_1 ,x_2

$$

two elements such that $$

x_2 \notin \left\langle {x_1 } \right\rangle \,

$$

We know that

$$

\left| {HK} \right| = \frac{{\left| H \right|\left| K \right|}}

{{\left| {H \cap K} \right|}}

$$

In this case, since both have order p prime, the intersection must divide the order of the group, so $$

H \cap K = \left\{ e \right\}

$$

Then $$

\left\langle {x_1 ,x_2 } \right\rangle

$$

has order $ p^2 $ and is not cyclic.

Edited: thanks to groops

In some proofs of Sylow’s Theorems, one proves that for any prime power dividing the order of a group, there is a subgroup or order equal to that prime power. Sometimes one proves Sylow first, and then shows that any sylow p group has subgroups of all lower prime powers. Other times, (such as in Herstein’s book, my undergraduate book), one proves the more general result and has Sylow’s first theorem as a corollary.

With respect to the vague language you were referring to, it sounds so me like they were using group actions on a quotient group. They likely knew it had order p because a group of prime power has a nontrivial center, meaning that it either has order $p^2$ here or $p$. In one case you’re done, in the other we have this.

It could also be an application of the isomorphism theorems. They let one bounce back and forth between quotient groups, and if one is witty about prime factors, one can sort of ‘pull out’ prime.

In the end, it sounds like some use of the following facts: the isomorphism theorems, Lagranges Theorem, the center of a prime powered group is nontrivial, and the class equation.

- Show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$
- Why is the Euclidean metric the natural choice?
- Prove $\frac{|a+b|}{1+|a+b|}<\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}$.
- Notation for sequences
- Inverse of a matrix is expressible as a polynomial?
- About the existence of the diagonal set of Cantor
- Is there a function with the property $f(n)=f^{(n)}(0)$?
- What's the difference between 'any', 'all' and 'some'?
- Prove that $(0,1)$ is cardinally equivalent to $[0,1)$
- How to calculate $\lim\limits_{{\rho}\rightarrow 0^+}\frac{\log{(1-(a^{-\rho}+b^{-\rho}-(ab)^{-\rho}))}}{\log{\rho}} $ with $a>1$ and $b>1$?
- Axiomatizing oriented cobordism
- Rudin Theorem 2.41 – Heine-Borel Theorem
- A proof of the Isoperimetric Inequality – how does it work?
- Conservation Laws: Difference and Reasonability Weak solutions, Integral solutions, distributional solutions
- Division algorithm for multivariate polynomials?