Articles of group theory

If $F(a, b)=\langle a, B\rangle$ then $B=a^ib^{\epsilon}a^j$: a neat proof?

If you take a generating pair for $F(a, b)$, $(a, B)$, then it is intuitively obvious that $B=a^ib^{\epsilon}a^j$ for $i, j\in \mathbb{Z}$, $\epsilon=\pm 1$. However, I cannot come up with a neat proof of this, and so was wondering if someone could either provide a reference or come up with a more elegant approach. My […]

Prove that G is a cyclic group

Suppose that $|G| = pq$ where $p$ and $q$ are primes such that $p < q$ and $p$ does not divide $q − 1$. Prove that $G$ is a cyclic group. A cyclic group is a group that has a unique generator element, so is the way to go with this to find that element? […]

If $G$ is isomorphic to all non-trivial cyclic subgroups, prove that $G\cong \mathbb{Z}$ or $G\cong \mathbb{Z}_p$

Assuming $G$ is isomorphic to all its non-trivial subgroups, prove that $G$ is isomoprhic to either $\mathbb Z$ or $\mathbb Z_p$ for $p$ prime.

How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

This question already has an answer here: Determining the number of subgroups of $\Bbb Z_{14} \oplus \Bbb Z_{6}$ 1 answer

Suppose that half of the elements of G have order 2 and the other half form a subgroup H of order n. Prove that H is an abelian subgroup of G.

Let $n>1$ be a positive integer. Let $G$ be a group of order $2n$. Suppose that half of the elements of G have order 2 and the other half forma a subgroup H of oder n. Prove that H is an abelian subgroup of G. I can only deduce from the index of H that […]

Classify groups of order 6 by analyzing the following three cases:

(i) G contains an element of order 6. (ii) G contains an element of order 3 but none of order 6. (iii) All elements of G have order 1 or 2. I’ve got: (i) $G$ is the cyclic group of order 6, isomorphic to $\mathbb{Z}_6^+$. (ii) $G$ is $S_3$, the group of permutations of 3 […]

A question on a generalization of perfect numbers

First of all, I would like to call a group immaculate provided that the orders of $G$ and $\Sigma$ (the order of $N$) where $N$ varies over all normal subgroups of $G$, are equal. From here it has been said that any abelianizer of a group whose order is less than that of $D(G)$, where […]

Is the “binary operation” in the definition of a group always deterministic?

The introduction to group theory that I’m reading requires that the actions of a group are “deterministic”; but the formal definition given makes no mention of this property: A set G is a group if the following criteria are satisfied. There is a binary operation $\cdot$ on $G$. That operation is associative… There is an […]

What are the length of the longest element in a Coexter group for every type?

What are the length of the longest element in a Coxeter group for every type? Thank you very much.

How to prove this permutation summation is zero.

The function $\mu_j$ characterises the position of the $j^\text{th}$-particle, where $$\mu_j = \left(j – \frac{N+1}{2}\right)d,$$ and $N$ is the total number of particles. I want calculate this summation: $$\sum_\sigma \left[\sum_{j=1}^N \mu_j \mu_{\sigma(j)}\right],$$ where $\sigma$ ranges over elements of the permutation group on $N$ objects. Implementation in Mathematica finds that this evaluates to zero. How can […]