# 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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### Proof that Möbius transformations are group under composition - finding inverse element

The task given in my textbook was to find which algebraic structure is $(X, *)$, where $X$ is set of Möbius transformations $x\rightarrow y=\frac{ax+b}{cx+d}$ in $\mathbb R$ and $*$ is composition. I ...
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### How to determine the order of the group $\langle a,b,c |a^2=b^2=c^2=(ab)^2=(bc)^4=(ca)^4=1 \rangle$?

How to determine the order of the group $\langle a,b,c |a^2=b^2=c^2=(ab)^2=(bc)^4=(ca)^4=1 \rangle$ ? I have almost no idea how to go on about this . Please help . Thanks in advance
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### Is the convolution algebra a *-algebra?

Let $G$ be a finite abelian group with $n$ elements. Consider the convolution algebra $C^*(G) \subset l^2(G)$, with multiplication: $$(a * b )(g) = \frac{1}{n}\sum_{x\in G} a(x)b(g-x)$$ Is there a ...
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### Infinite groups with all elements of order 2?

If G is a group such that $a^2 =e$ for all $a \in G$, where $e$ is the identity element in $G$, then $G$ is finite. This question can be proved false if we can get a group of infinite order with ...
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### prime-power order subgroups of a quotient group

Let $G$ be a finite group. Suppose that $M$ is normal $p$-subgroup of $G$. Then every prime-power order subgroup of $G/M$ is either a $p$-subgroup or a $q$-subgroup of the form $QM/M$ for some $q$-...
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### Groups and rings of order $p^2$.

Up to isomorphism there are exactly two abelian groups of order $p^2$. there are exactly two groups of order $p^2$. there are exactly two commutative rings of order $p^2$. there is exactly one ...
Question: Let G be a group and let $g \in G$. If $z \in Z\left ( G \right )$, show that the inner automorphism induced by g is the same as the inner automorphism induced by $zg$. That is show that,...
### subnormal subgroup of a $p$-group
I could not reason with this statement of a proof I am currently looking at. Let $G$ be a finite $p$-group with $Q\leq G$ and suppose that $H$ is a cyclic subgroup of $Q$ then $H$ is subnormal in \$...