# Tagged Questions

Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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### A Finite Group of Nonprime Order which is Unique up to Cyclic Group

Maybe I have to wait until I learn and study more, but I just became curious. I know that every finite group of prime order is cyclic, and hence unique up to isomorphism. I have 2 questions about this....
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### Factorizing elements of a group into a product of generators.

$$s = (1\ 2) \\ t = (1\ 2\ 3\ ...\ n)$$ Given the Symmetric Group $S_n$ generated by $s$ and $t$, is there a way to quickly factor an element $g \in S_n$ into a minimal product of positive powers ...
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### Prove that $S= \langle(1,2,3,4) \rangle$ has 3 conjugates in $S_4$

Some things I know: $S = \{ (1),(1,3)(2,4), (1,2,3,4),(1,4,3,2)\}$ $(2,4) \in N_G(S)$ Number of conjugates = $[G: N_G(S)]$ This seems like such a easy question but it made me realised that I do ...
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### Reconciling Different Definitions of Solvable Group

Looks like my class note defines solvable group differently from others: A finite group $G$ is called solvable if $H' \neq H$ for each subgroup $H$ of $G$ different from $\{1\}$, where $H'$ is ...
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### Showing that $\{0, 1\}$ is a group under addition modulo $2$

I'm considering a set $G = \{0,1\}$ under addition modulo 2. I.e. $$a*b = a + b\bmod 2, \quad \quad \forall \ a,b \in G.$$ I am able to show that there exists an identity element, $0$. Showing ...
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### on the characters of the normal subgroup and its quotient

I read character theory recently and thought about the following proposition, but I do not know is this true or false: Let $G$ be a finite group such that two distinct primes $p$ and $q$ ...
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### Finite Group is Subgroup of Its Radical's Automorphism

I am still working on this problem on radical of finite group: Assume that $R(G)$ is simple and not commutative, show that $G$ is a subgroup of $Aut(R(G))$. I have managed to parse the problem ...
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### Find a composition series of $D_8$

Answer: Let $D_8=\langle a,b\rangle=\{e,a,a^2,a^3,b,ba,ba^2,ba^3\}$, where $a^4 = b^2 = e$ and $ab = ba^{-1}$ as usual. Then $$\{e\}\lhd\langle a^2\rangle\lhd\langle a\rangle\lhd D_8$$ or \{e\}\...
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### $|G|=24$ prove that $G$ is not simple
Let $G$ be a group of order 24, and we shall assume there there exist a non-normal 2-sylow group in $G$. i want to show that it's not simple. first i have showed that there are exactly three 2-sylow ...
### There is no simple group of order $144$
There is no simple group of order $144$ I have a question to the proof of the statement above (from the book J. Gallian, Contemporary abstract algebra), it is about the index theorem, so I give ...