# 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 order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?

Given $|G|=p^2$ then how can you deduce $G\cong C_{p^2}$ or $G \cong C_p \times C_p$ I have shown that G is abelian, not sure what to do next
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### solve the exercise in an alternative way

I have this exercise: Assume that G and H are finite groups and that $|G|$ and $|H|$ are relatively prime. Show that the only group-homomorphism $\phi: G \rightarrow H$ is the trivial one. I want to ...
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### Orbits of $2 \times 2$ matrices over $\Bbb F_2$

Let $X$ be the set of $2 \times 2$ matrices over field $\Bbb F_2$, and $G \subset X$ be the group of invertible $2 \times 2$ matrices over $\Bbb F_2$ Let $G$ act on $X$ by $g*x = gxg^{-1}$ Need to ...
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### Parity of permutation example

I know the definition of parity of permutation. But what does that look like in examples? For example, if the number of permutations is odd, then the sign of permutation in $-1$. What does this mean? ...
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### Number of non-isomorphic non-abelian groups of order 10

Number of non-isomorphic non-abelian groups of order 10 Let $G$ be a group of order 10.Let $a\neq e \in G$ then it is not possible that all elements are of order 2 otherwise $G$ becomes abelian. ...
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### Suppose $G$ is a group with $|G|=35$. Prove that if $H$ is a subgroup of $G$ with order 7, then $H$ is a normal subgroup of $G.$

(1) Suppose $G$ is a group with $|G|=35$. Prove that if $H$ is a subgroup of $G$ with order $7$, then $H$ is a normal subgroup of $G.$ (2) Suppose that G is a group with $|G|=35.$ Prove that if $G$ ...
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### Given a group is finite and non-abelian, why is the left coset with the centre of the group non-cyclic?

Assume $T$ is finite and non-abelian then why is $T/Z(T)$ non-cyclic? Where $Z(T)$ is the centre of the group $T$. I've shown $Z(T)$ is a normal subgroup of T, but not sure what to do next or if ...
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### Computing the order, inverse, and parity of a permutation

How do you compute the order, inverse and parity of $\alpha=(12)(43)(13542)(15)(13)(23)$? Please explain all steps taken to get the answer. I guess my thought process was to first put it into a ...
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### Way to show that there exists $n\neq 0$ such that $n^{(p-1)/2}\neq 1 \mod p$

Suppose that $p\ge3$ is a prime and that $0\neq n \in \mathbb{Z}/p\mathbb{Z}$. Suppose also that we want to show that there is an $n$ such that $n^{(p-1)/2} \neq 1 \mod p$. One way to show this would ...
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### Coset to a power

my book states that if $N \trianglelefteq G$ then $(gN)^\alpha = g^\alpha N$ for $\alpha \in \mathbb{Z}$ The proof should be simple by induction but I can't understand how $(gN)^0 = [N = g^0N]$. How ...
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### Every subgroup of $(\mathbb {Z_n},+)$ is closed under multiplication

I am stuck in this proof that every subgroup of $(\mathbb {Z_n},+)$ is also a subring. which requires me to prove it is closed under multiplication. I have to show if $a,b \in G<\mathbb {Z_n}$ , ...
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### Homomorphisms $h:C_5 \to Aut(C_{31})$ (first course in group theory)

I am asked to show explicitly all homomorphisms $h: C_q \to Aut(C_{31})$ for $q=5,3,2,29$ For $q=5$, $C_5 =\{1,y^2,...,y^4\}$ and $Aut(C_{31})=C_{30}=\{1,\alpha,...,\alpha^{29}\}$ so we know the ...
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### An Example for a Graph with the Quaternion Group as Automorphism Group

I am reading "Graphs of Degree Three with given Abstract Group" (by Robert Frucht) where the author describes (somewhat tedious) algorithms to construct suitable graphs starting from a given group. I ...
G is a finite group, $\phi:G \to G$ a homomorphism. $\psi:G \to G$ is a homomorphism defined by $\psi(x)=\phi(\phi(x))$. Prove that $(\ker\phi= \ker \psi)\implies($Im$\psi=$Im$\phi)$. Can someone ...