# Tagged Questions

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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### If $|G|=p^3q^2$ then $\Phi(G)$ is cyclic for primes $p\neq q$.

I have conjectured this result for the Frattini subgroup by doing some calculations in GAP. I think this is even true if $|G|=p_1^{i_1}\cdots p_n^{i_n}$ for $i_j\leq 3$ holds, but I would like to ...
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### Kazhdan's property (T) vs residual finiteness

There is a theorem that states that a discrete group $G$ with Kazhdan's Property $(T)$ and Property $(F)$ (so called factorisation property) is residually finite (see Kirchberg, Discrete groups with ...
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### Minimal Normal Subgroups of an elementary abelian p-group

Can you explain/prove, why the number of minimal normal subgroups of an elementary abelian $p$-group of order $p^n$ (for instance of $\mathbb{Z_p}^n$), is exactly $(p^n-1)/(p-1)$? I know that it seems ...
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### Show that $\phi(N) \leq H$

Let $\phi: G \to H$ be a group homomorphism and $N \leq G$ (with $G$, $N$ and $H$ groups). Show that $\phi(N) \leq H$ So this is what I did: Obviously $\phi(N)$ is a subset of H because $N$ is a ...
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### The Cardinality of a subset of field of 8 elements

Let $F$ be a field of $8$ elements. Let $A$ be a subset of $F$ and $A = \{x \in F \mid x^7=1$ and $x^k \neq 1$ for $k<7\}$. Then find the number of elements in $A$. Argument: Since $F$ is a ...
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### if HK is a subgroup for all K, does it imply that H is normal?

we know that in a group G, if H,K be subgroups such that H is normal, then the product HK is also a subgroup. does the converse hold? i.e. if H is a subgroup of a group G such that for any subgroup K ...
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### Cayley table of a group of order n with 2n entries deleted

Let $G$ be a group of order $n$, on elements $x_1, ..., x_n$. Let's draw the Cayley table of the group, that is, table $A$ with $a_{i,j}$ = [number of $x_i \circ x_j$ in the list $x_1, ..., x_n$] ...
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### Group such that Aut(G) = H and Aut(H) = G

So this question just popped into my mind: does there exist a group $G$ such that $Aut(G) = H$ and $Aut(H) = G$ and $H\not=G$. My intuition says no, but im not sure how to go about proving this. Can ...
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### For which numbers $n$ does every group with order $n$ have a non-trivial center?

Which numbers $n$ have the property that every group with order $n$ has a non-trivial center ? $n$ has this property if it is an abelian number (every group with order $n$ is abelian). If $n$ is a ...
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### Finite Fields and Discrete Log Problem

The finite field $\mathbb F_{p^p}$, for a prime $p$, can be described as $\mathbb F_p[x]/(x^p−x−1)$. Let $G=\mathbb F_{p^p}^\times/\mathbb F_{p}^\times$. Estimate the cost of obtaining the discrete ...
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### $\otimes$-Categorical Generalization of Lagrange

I am reading M. Brandenburg's paper and came across the following result which is a generalization of Lagrange's theory in group theory: Let $\mathcal C$ be a $\otimes$-category and $A\to B$ a ...
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### Elementary consequences of commuting limits and colimits over groups

In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is mentioned: Theorem 1. $H$-limits commute ...
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### Symmetry group of product polytope

The symmetry group of the interval $[-1,1]$ is $\mathbb Z_2$, since it consists only of the identity and the reflection at the origin. Consider now the square $[-1,1]^2$. Obviously, its symmetry ...
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### Question on special unitary group $SU(3)$ over local $p$-adic field

I am reading Casselman and Shalika's paper, the unramified principla series of p-adic groups II. I have a problem on the special unitary group $SU(3)$ over local p-adic field. Let $F$ be a local ...
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### Is the free product $\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}$ a linear group?

Let $H:=\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}=\langle p,q| q^n=1\rangle.$ I want to know if $H$ is a ($\mathbb{Z}$)linear group that is to say is there an injective homomorphism $f: H\to GL_m(\mathbb{Z})$ ...
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### Infinite group with unique maximal subgroup

If $G$ is a group, then by a maximal subgroup, we mean a proper subgroup, which is maximal w.r.t. subset(subgroup) relation. It is well known that a finite group with unique maximal subgroup must be ...
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### Why $K \cap H$ is a maximal subgroup of $H$?

Suppose $G$ is a finite group and $H \unlhd G$. Suppose that $H \cap M$ is either $H$ or a maximal subgroup of $H$ for any maximal subgroup $M$ of $G$. Let $N$ be a minimal normal ...
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### Infinitely iterated square roots in groups

Let $G$ be a group. What are possible conditions on $G$ to ensure that there is no sequence $\{g_i\}_{i\in\mathbb Z}\subset G\backslash\{1\}$ such that $g_{i+1}=g_i^2$ for all $i\in\mathbb Z$? Does ...
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What I am about to ask is related to the question presented here. Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, define$$X^H = \{x : hx = x \text{ for all }h \... 0answers 39 views ### Proving a version of the Kronecker's Theorem I am trying to prove the following version of the Kronecker's Theorem: Suppose k is a positive integer and \{1, \theta_0, \dots, \theta_{k-1}\} is linearly independent over \mathbb Q. Then \... 0answers 64 views ### Haar measure on a profinite group is the inverse limit of the counting measure on its quotients? I've heard this a few times now, though I've never seen a precise result. I guess the precise statement would be close to: Let N_i be a basis normal subgroup neighborhoods of the identity in a ... 0answers 63 views ### Is the monster group a characteristic quotient of F_2? Let F_2 be the free group on two generators, and M the monster group. It's known that every finite simple group is 2-generated, so let F_2\rightarrow M be a surjection with kernel N. Let ... 0answers 49 views ### Finite conjugacy classes in a certain group with three generators Def. We say that group G has non-trivial finite conjugacy class if there is a conjugacy class C=\lbrace g_i \rbrace such that g_i \neq 1 of G with |C|<\infty. Let G be group$$<a,b,c \...
Let $G$ be a group for which the center $Z(G)$ is of index $n$. How to prove that an element of the derived subgroup $G^\prime$ is the product of at most $n^3$ commutators?
Let $F$ be a free group and let $V$ be another verbal subgroup of $F$ such that $$F = [F,F] V.$$ Is it true that $V=F?$ More generally, if $N$ is a normal (or even characteristic) subgroup of $F$ ...