## Groups acting on themselves by left multiplication-Cayley’s theorem[Problems in the section 4.2 of ”Dummit and Foote: Abstract Algebra”]

These are  problems,  Its solutions will coming soon! Pdf file 42_problems

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Solutions 42_solutions

## Group actions and permutation representations[Problems in the section 4.1 of ”Dummit and Foote: Abtract Algebra”]

Let $G$ be a group and let $A$ be a nonempty set.

1. Let $G$ act on the set $A$. Prove that if $a,b\in A$ and $b=g\cdot a$ for some $g\in G$, then $G_b=gG_ag^{-1}$($G_a$ is the stabilizer of $a$). Deduce that if $G$ acts transitively on $A$ then the kernel of the action is $\cap_{g\in G}gG_ag^{-1}$.

2. Let $G$ be a permutation group on the set $A$(i.e., $G\leq S_A$), let $\delta \in G$ and let $a\in A$. Prove that $\delta G_a\delta^{-1}=G_{\delta(a)}$. Deduce that if $G$ acts transitively on $A$ then $\cap_{\delta \in G}\delta G_a\delta^{-1}=1.$

3. Assume that $G$ is an abelian, transitive supgroup of $S_A$. Show that $\delta (a)\not =a\forall \delta\in G-\{1\}\forall a\in A$. Deduce that $|G|=|A|$[Use the preceding exercise.]

4. Let $S_3$ act on the set $\Omega$ of ordered pairs: $\{(i,j)|1\leq i,j\leq 3\}$ by $\delta ((i,j))=(\delta (i),\delta (j))$. Find the orbits of $S_3$ on $\Omega$. For each $\delta \in S_3$ find the cycle decomposition of $\delta$ under this action (i.e., find its cycle decomposition when $\delta$ is considered as an element of $S_9$– first fix a labelling of these nine ordered pairs). For each orbit $\mathcal{O}$ of $S_3$ acting on these nine points pick some $a\in\mathcal{O}$ and find the stabilizer of $a$ in $S_3$.

5. For each parts (a) and (b) repeat the preceding exercise but with $S_3$ action on the specified set:

(a)The set of $27$ triples $\{(i,j,k)|1\leq i,j,k\leq 3\}$

(b)The set $\mathcal{P}(\{1,2,3\})-\{\emptyset\}$ of all $7$ nonempty subsets of $\{1,2,3\}$.

6. Let $R$ be the set of all polynomials with integer coefficients in the independent variables $x_1,x_2,x_3,x_4$ and $S_4$ act on $R$ by permuting the indices of the four variables:$\sigma\cdot p(x_1,x_2,x_3,x_4)=p(x_{\sigma (1)},x_{\sigma (2)},x_{\sigma (3)},x_{\sigma (4)})$ for all $\sigma \in S_4$ and $p\in R$.

a)Find the polynomials in the orbit of $S_4$ on $R$ containing $x_1+x_2$;

b)Find the polynomials in the orbit of $S_4$ on $R$ containing $x_1x_2+x_3x_4$;

c)Find the polynomials in the orbit of $S_4$ on $R$ containing $(x_1+x_2)(x_3+x_4)$.

7. Let $G$ be a transitive permutation group on the finite set $A$. A block is a nonempty subset $B$ of $A$ such that for all $\sigma \in G$ either $\sigma (B)=B$ or $\sigma (B)\cap B=\emptyset$.

a)Prove that if $B$ is a block containing the element $a$ of $A$ then $G_B:=\{\sigma \in G|\sigma (B)=B\}$ is a subgroup of $G$ containing $G_a$;

b)Show that if $B$ is a block and $\sigma_1(B),\cdots,\sigma_n(B)$ are all dinstinct images of $B$ under the elements of $G$ then these form a partion of $A$;

c)A transitive group $G$ on a set $A$ is said to be primitive if the only blocks in $A$ are the trivial ones: the sets of size $1$ and $A$ itself. Show that $S_4$ is primitive on $A=\{1,2,3,4\}$. Show that $D_8$ is not primitive as a permutation group on the four vertices of a square;

d)Prove that the transitive group is primitive of $A$ iff for each $a\in A$, the only subgroups of $G$ containing $G_a$ are $G_a$ and $G$.

8. A transitive permutation group $G$ on a set $A$ is called doubly transitive if for any (hence all) $a\in A$ the subgroup $G_a$ is transitive on the set $A-\{a\}$.

a)Prove that $S_n$ is doubly transitive on $\{1,2,\cdots,n\}$ for all $n>1$;

b)Prove that a doubly transitive group is primitive. Deduce that $D_8$ is not doubly transitive in its action on the $4$ vertices of a square.

9. Assume $G$ acts transitively on the finite set $A$ and let $H$ be a normal subgroup of $G$. Let $\mathcal{O}_1,\mathcal{O}_2,\cdots,\mathcal{O}_r$ be the distinct orbits of $H$ on $A$.

a)Prove that $G$ permutes the sets $\mathcal{O}_i$. Prove that $G$ is transitive on $\{\mathcal{O}_i\}$. Deduce that all orbits of $H$ on $A$ have the same cardinality;

b)Prove that if $a\in\mathcal{O}_1$ then $|\mathcal{O}_1|=|H:H\cap G_a|$ and $r=|G:HG_a|$.

10. Let $H$ and $K$ be subgroups of the group $G$. For each $x\in G$ define the $HK$ double coset of $x$ in $G$ to be the set $HxK=\{hxk|h\in H, k\in K\}.$

a)Prove that $HxK$ is the union of the left cosets $x_iK$, where $\{x_iK\}$ is the orbit containing $xK$ of $H$ acting by left multiplication on the set of left cosets of $K$;

b)Prove that $HxK$ is the union of right cosets of $H$;

c)Prove that $HxK$ and $HyK$ are either the same set or are disjoint for all $x,y\in G$. Show that the set of $HK$ double cosets partitions $G$;

d)Prove that $|HxK|=|K|\cdot |H:H\cap xKx^{-1}|$;

e)Prove that $|HxK|=|H|\cdot |K:K\cap x^{-1}Hx|$.

P.S. These problems are from ”Dummit and Foote, Abstract Algebra”. Solutions will coming soon! 😀

Pdf file: 41_problems

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Solutions:41_solutions