# Tagged Questions

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### Abstract Monomorphism 3 part Question

I have been working on this problem for an hour now and gotten nowhere: Let $G$ be any group and $A(G)$ the set of all 1-1 mappings of $G$, as a set, onto itself. Define $L_a : G \rightarrow G$ by ...
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### Group's morphisms

I know that the constant map equal to $1$ and the signature are two group's morphisms from the group of permutations $(\mathcal S_n,\circ)$ to the group $(\Bbb C^*,\times)$. My question is: Can we ...
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### An epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$

I want an explicit example of an epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$. Thanks.
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### Normal subgroups, direct product and monomorphism problem

Let $G$ be a group and let$H,K$ be normal subgroups of $G$. Let $\pi_H,\pi_K$ be the projections on $H$ and $K$ respectively. Show that the map $$f:G/(H \cap K) \to G/H \times G/K$$ defined as ...
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### Isomorphism between quotient groups

Exercise Let $f:G \to G'$ be an isomorphism and let $H\unlhd G$. If $H'=f(H)$, prove that $G/H \cong G'/H'$. As I've shown that $H'\unlhd G'$, I thought of defining $$\phi(Ha)=H'f(a)$$I was trying ...
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### Order of $M(x)$, where $M\colon G\to H$ is an injective homomorphism

Let $M\colon G\to H$ be a homomorphism and let $x$ be in $G$. Suppose that $x$ has order $k$. Show that if $M$ is injective the order of $M(x)$ equals order of $x$. My approach: ...
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Given a group $G$, we can define its central automorphism group by $$\operatorname{Aut}_c(G)= C_{\operatorname{Aut}(G)}(\operatorname{Inn}(G)) = \{ \phi\in\operatorname{Aut}(G) : \phi(g)g^{-1} \in ... 2answers 115 views ### How to show there is only one homomorphism between \mathbb Z_{25} and S_4? Show that there is only one homomorphism from \mathbb Z_{25} to S_4. How do i approach this question? If anyone could give me a hint or some guidelines that would be great. 3answers 130 views ### Question about automorphisms Let G be a finite abelian group and let n be a positive integer relatively prime to |G|. a. Show that the mapping ϕ(x)=x^n is an automorphism of G. b. Show that every x∈G has an nth root, i.e., for ... 2answers 45 views ### group,subgroup and isomorphism I study group theory now but I could not understand isomorphisms very well. In the book that I study I have seen that; \mathbb{Z}_6=\{0,1,2,3,4,5\} is given and H=\{0,2\} is a subgroup of the ... 1answer 149 views ### Automorphism of \mathbb{Z}\rtimes\mathbb{Z} I'm looking for a description of \mathrm{Aut}(\mathbb{Z}\rtimes\mathbb{Z}). I've tried an unsuccessfully combinatorical approac, does anymore have some hints? Thank you. 1answer 64 views ### number of automorphisms for group in order 169 Let G be a group with order 169. Prove number of automorphisms is at least 143. I thought that 169 is 13 squared so maybe G isomorphic to  Z_{169}  but I dont have any idea. How can I solve ... 1answer 92 views ### Hypercube and dihedral group Let G_n denote the subgroup of the orthogonal group O_n of elements that send the hypercube to itself, the group of symmetries C_n, including the orientation-reversing symmetries. It would ... 0answers 37 views ### Homomorphism on the group of isometries Prove that the map f: M \rightarrow \{1,r\} defined by t_a \rho_{\theta} \mapsto 1, t_a \rho_{\theta}r \mapsto r is a homomorphism. M denotes the set of isometries of the plane; r the reflexion ... 1answer 192 views ### Problem on abelian group Let G be an abelian group, and \Phi:G\to \mathbb{R} is a function with the following property:$$\forall a,b\in G,~~ |\Phi(a+b)-\Phi(a)-\Phi(b)|<c$$The problem asks to prove the existence of ... 0answers 88 views ### Number of kernals of all \mathbb{Z}_n to \mathbb{Z}_m homomorphisms? How many subgroups K \le \mathbb{Z}_n are there with K =\ker(\phi) for some homomorphism \phi\colon\mathbb{Z}_n \rightarrow \mathbb{Z}_m? Stuck and need a hint. Have so far that there are ... 2answers 403 views ### Prove the third isomorphism theorem I'm trying to prove the third Isomorphism theorem as stated below Theorem. Let G be a group, K and N are normal subgroups of G with K⊆N. Then$$(G⁄K)⁄(N⁄K)≅G⁄N.$$I look up for some ... 3answers 147 views ### Aut \mathbb Z_p\simeq \mathbb Z_{p-1} I'm trying to prove that Aut \mathbb Z_p\simeq \mathbb Z_{p-1} (p prime). I know that Aut \mathbb Z_p has p-1 elements because \mathbb Z_p has p-1 possiblities of generators, so intuitively ... 1answer 85 views ### Prove that Tx=x^{-1}~\forall~x\in G. [duplicate] Let G be a finite group and suppose the automorphism T sends more than \dfrac{3}{4}th of the elements of G onto their inverses. Prove that Tx=x^{-1}~\forall~x\in G. 2answers 627 views ### If \phi:G\to\bar{G} is an isomorphism and if H is a normal subgroup of G, then \phi(H) is a normal subgroup of \bar{G}. If \phi:G\to\bar{G} is an isomorphism and if H is a normal subgroup of G, then \phi(H) is a normal subgroup of \bar{G}. I am struggling with getting started with the problem. I know ... 1answer 123 views ### How I can find the inverse of an isomorphism? The motivation of this question can be found in Can we extend the map φ to ℝ^{r}×C(ℚ)^{\text{tors}}→C(ℚ) as an isomorphism or not? My question is: How I can find the inverse of the ... 0answers 126 views ### Order of kernel of a homomorphism Let p,q be distinct primes. Prove that the kernel of the map$$f: (\mathbb{Z}/p^k\mathbb{Z})^* \rightarrow (\mathbb{Z}/p^k\mathbb{Z})^*$$defined by f(x)=x^q has order \gcd(p-1,q). Thank ... 1answer 323 views ### Question on a homomorphism of a set G. I'm having difficulty showing the given a map, say \phi(z)=z^k, is surjective. This question is from D & F section 1.6 - #19 Let G =\{z \in \mathbb C|z^n=1 \text{ for some } n \in \mathbb ... 3answers 137 views ### How can I conclude that an automorphism is the identity map? Let \phi be an automorphism on a group G. If \phi maps any one non-identity element in G to itself, is \phi necessarily the identity map? What if G is cyclic? 5answers 185 views ### Show that S_2 is isomorphic to Z_2 Show that S_2 is isomorphic to Z_2 I know that S_2 has two elements \sigma_1 and \sigma_2...and Z_2 has two elements 0 and 1. But isomorphism is a bijective function, right? So doesn't ... 1answer 140 views ### On automorphisms group of order p^n Let G be a finite group, such that \mid Aut(G)\mid=p^n. Then prove G is p-group or G\cong P\times C_{2}, where P is a p-group. Thank you 1answer 114 views ### On automorphisms group C_{2}\times D_{8} Let D_{8} be group dihedral of order 8 and C_{2} be cyclic group of order 2. Then determine the number all automorphisms of C_{2}\times D_{8}. Can you determine automorphisms group of ... 2answers 552 views ### Is every group the automorphism group of a group? Suppose G is a group. Does there always exist a group H, such that \operatorname{Aut}(H)=G, i. e. such that G is the automorphism group of H? EDIT: It has been pointed out that the answer ... 0answers 161 views ### Homomorphisms and Automorphisms between cyclic groups of prime order Let A = B \times C where B and C are cyclic of order p and p^2 respectively, where p a prime. How many endomorphisms are there? How many of these endomorphisms are automorphisms? 1answer 128 views ### Homomorphisms between sets of homomorphisms Let A,B be finite, commutative groups. Let A^{*} = Hom(A, \mathbb{Q}/\mathbb{Z}), the set of homomorphisms from A to \mathbb{Q}/\mathbb{Z}. A^{*} is abelian itself (take this for granted). Let ... 2answers 4k views ### Can we ascertain that there exists an epimorphism G\rightarrow H? Let G,H be finite groups. Suppose we have an epimorphism$$G\times G\rightarrow H\times H Can we find an epimorphism $G\rightarrow H$?
Let $f : G \to G'$ be a group morphism. I need to find a necessary and sufficient condition such that $\operatorname{Im}(f)$ is a normal subgroup of $G'$.