Articles of group theory

Any two groups of three elements are isomorphic – Fraleigh p. 47 4.25(b)

The answer has no details. Hence maybe the answer is supposed to be quick. But I can’t see it? Hence I took two groups. Call them $G_1 = \{a, b, c\}, G_2 = \{d, e, f\}$. Then because every group has an identity, I know $G_1, G_2$ has one each. Hence WLOG pick $c$ as […]

Showing that a homomorphism between groups of units is surjective.

This question already has an answer here: Why does the natural ring homomorphism induce a surjective group homomorphism of units? 2 answers

Character on conjugacy classes

Let $V_j$, $j = 1,2$ be finite dimensional representations of a group $G$. Show: $\chi_{V_j}$ is a constant on each conjugacy class of $G$, where $\chi_{V_j}$ is the character of the representation. I’ve just started with group theory and have a really hard time so I’d like someone to confirm what I did so far […]

Commutator Identities in Groups

Let $x$, $y$, $z$ be elements of a group $G$ and let $[x,y]=x^{-1}y^{-1}xy$ be the commutator of $x$ and $y$. Then we have the following identities: $[x,zy]=[x,y][x,z][[x,z],y]$ $[xz,y]=[x,y][[x,y],z][z,y]$ My question is, is there any identity for $[x_1x_2\cdots x_m, y_1y_2\cdots y_n]$, generating the identities above? It may not be difficult but I did not find it […]

Confusion about the hidden subgroup formulation of graph isomorphism

I am going through Quantum factoring, discrete logarithms and the hidden subgroup problem by Richard Jozsa. On page 13, the author discussed the hidden subgroup problem (HSP) formulation of the graph isomorphism (GI) problem. I would like to make it sure that I get the development of the concept right. Here both $A$ and $B$ […]

'Randomness' of inverses of $(\mathbb{Z}/p \mathbb{Z})^\times$

Suppose you are given the group $(\mathbb{Z} / p \mathbb{Z})^{\times}$, where $p$ is prime. Let $A_p$ denote the sequence whose $j$th element is the inverse of $[j]$. For instance, if $p = 7$, the sequence $A_7$ is $(1,4,5,2,3,6)$. Suppose you truncate the sequence upto the $\alpha p$th term (where $\alpha$ is a very small constant […]

The following groups are the same.

This is an exercise from J.J.Rotman’s book: Prove that the following groups are all isomorphic: $$G_1=\frac{\mathbb R}{\mathbb Z},G_2=\prod_p{\mathbb Z(p^{\infty})}, G_3=\mathbb R\oplus\big(\frac{\mathbb Q}{\mathbb Z}\big)$$ What I have done is: Since $tG_1=\frac{\mathbb Q}{\mathbb Z}$, which $t$ means the torsion subgroup; and the fact that $G_1\cong tG_1\oplus\frac{G_1}{tG_1}$ so I should show that $\frac{\mathbb R}{\mathbb Q}\cong\mathbb R$. A theorem […]

why the column sums of character table are integers?

There is a well-known result of Solomon which states that sum of entries of any row in $\mathbb{C}$-character table of a group $G$ is an integer number. It is mentioned in Martin Isaacs Character Theory of Finite Group as a note that the column sums are also integers. My question is that what’s the reason […]

Determine the center of the dihedral group of order 12

Determine the center of the dihedral group of order 12. This was asked in an exam so I presume there must be a more efficient way of doing it than actually going through all the elements of a group G and checking that they commute with every other element. When searching for an answer online […]

$\mathbb Z_p^*$ is a group.

I’m trying to prove that $\mathbb Z_p^*$ ($p$ prime) is a group using the Fermat’s little theorem to show that every element is invertible. Thus using the Fermat’s little theorem, for each $a\in Z_p^*$, we have $a^{p-1}\equiv1$ (mod p). The problem is to prove that p-1 is the least positive integer which $a^{p-1}\equiv1$ (mod p). […]