# 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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### Show that left cosets partition the group

I know how to prove that it happens, by proving that the left coset definition actually is an equivalence relation. Then, it's proved that it partitions the set, since equivalence relations do it. ...
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### How to generate the icosahedral groups $I$ and $I_h$?

The icosahedral groups $I$ with 60 elements and $I_h = I \times Z_2$ are also three dimensional point groups. However, ever unlike other point groups, it seems there is rarely reference to give their ...
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### Existence of isomorphism $\varphi:S_4\to \mathbb{Z}_8$

I need to prove or disprove: Existence of isomorphism $\varphi:S_4\to \mathbb{Z}_8$ My attempt: No, there isn't isomorphism, because if it did then $S_4$ would have an element of order $8$, ...
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### Number of elements of order $11$ in group of order $1331$

Let $G$ be a group of order $1331$. Prove that $G$ has at least $11$ elements of order $11$. $|G|=1331=11^3$ So by First Sylow's theorem, there exists a Sylow $11$-subgroup of G. By Third Sylow's ...
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### Cardinality of automorphism groups of groups of order $p^4$.

As far as I know there is no classification of the automorphism groups of groups of order $p^4$. (see http://mathoverflow.net/questions/157049/classification-of-automorphism-groups-of-groups-of-order-...
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### are these two finite groups with different presentation isomorphic?

Consider two groups $$\langle x,y \, | \, x^4=y^5=1 ,yxy=x \rangle$$ and $$\langle a,b \, | \, a^{10}=1,b^2=a^5,aba=b \rangle.$$ I think they are isomorphic, but I can't show it, it will be great if ...
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### Noncyclic Abelian Group of order 51

The problem is to prove or disprove that there is a noncyclic abelian group of order $51$. I don't think such a group exists. Here is a brief outline of my proof: Assume for a contradiction that ...
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### Computing Factor Group

I am reading John Fraleigh's First Course in Abstract Algebra, $\S$36 on the Second Isomorphism Theorem which says that if $H < G$ and $N \triangleleft G$, then $$(HN)/N \cong H/(H \cap N).$$ He ...
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### Permutation of cosets

Let $G$ be a finite group and $\gamma \in Sym(G)$, such that $\gamma (1) = 1$ and $\gamma (gH) = \gamma (g)H$ for all $g\in G$, $H\leq G$. This means $\gamma$ induces a permutation of the left cosets ...
### Are there groups of order $p^4q^2$ which are not semi-direct product?
It is easy to show that if $G$ is a group of order $p^2q^2$, where $p,q$ are primes with correspondings Sylow subgroups $P,Q$, that $G$ is a semi-direct product of $P$ and $Q$. Moreover, if $pq\neq 6$,...
### Verify that $A \oplus B$, where $A$ and $B$ are cyclic groups of orders 2 and 3, is the cyclic group of order 6
Let's define $A$ and $B$ as follows: $A$ = {e,a} $B$ = {e,b,2b} Then $A\oplus B= \{\{e+e\},\{e+b\},\{e+2b\},\{a+e\},\{a+b\},\{a+2b\}\}$ which is equal to \$\{\{e\},\{b\},\{2b\},\{a\},\{a+b\},\{a+2b\}\...