Groups Exercises I

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 9 February 2010

Groups Exercises I

1.

1. Show that the intersection of two subgroups of a group $G$ is a subgroup of $G.$
2. Give an example which shows that the union of two subgroups of a group $G$ need not be a subgroup of $G.$
3. Let $G$ be a group and let $H$ and $K$ be two subgroups of $G.$ Assume that $H⊈K$ and that $K⊈H.$ Show that $H\cup K$ is not a subgroup of $G.$

2. Let $G$ be a group and let $S$ be a subgroup of $G.$ Let $\mathscr{H}$ be the set of subgroups $H$ of $G$ such that $S\subseteq H.$ Define $$H_S=\bigcap _{H\in \mathscr{H}}H.$$

1. Show that, by the preceding exercise, $H_S$ is a subgroup of $G.$
2. Show that $S\subseteq H_S$ since $S\subseteq H$ for every $H\in \mathscr{H}.$
3. Show that if $H$ is a subgroup of $G$ and $S\subseteq H$ then $H\supseteq H_S.$

Conclude that $H_S=⟨S⟩.$ So $⟨S⟩$ is the smallest subgroup of $G$ containing $S.$

3. Lagrange's Theorem Let $G$ be a group and let $H$ and $K$ be subgroups of $G$ with $K\subseteq H\subseteq G.$ Show that $$\left|G:K\right|=\left|G:H\right|\left|H:K\right|.$$

Show that Corollary 2.3 is a special case of the theorem with $K=\left(1\right).$

4. Let $G$ be a group and let $H$ be a subgroup of $G.$ A double coset of $H$ in $G$ is a set $$HgH=\left\{hgh\text{'}|h,h\text{'}\in H\right\},$$ where $g\in G.$ The notation $H\backslash G/H$ denotes the set of double cosets of $H$ in $G.$

Show that the double cosets of $H$ in $G$ partition $G.$

5. Let $f:G\to H$ be a group homomorphism.

a) Let $M\subseteq G$ be a subgroup of $G$ and define $$f\left(M\right)=\left\{f\left(m\right)|m\in M\right\}.$$

1. Show that $f\left(M\right)$ is a subgroup of $H.$
2. Show that $f\left(M\right)\subseteq \mathrm{im}f=f\left(G\right).$
3. Show that if $f$ is surjective and $M$ is a normal subgroup of $G$ then $f\left(M\right)$ is a normal subgroup of $H.$
4. Gve an example of a homomorphism $f:G\to H$ and of a normal subgroup $M$ of $G$ such that $f\left(M\right)$ is not a normal subgroup of $H.$

$$\begin{array}{rcl}G& \stackrel{f}{\to }& H\\ M& \mapsto & f\left(M\right)\\ \cap |& & \cap |\\ G& \mapsto & f\left(G\right)=\mathrm{im}f.\end{array}$$

b) Let $N\subseteq H$ be a subgroup of $H$ and define $$f^{-1}\left(N\right)=\left\{g\in G|f\left(g\right)\in N\right\}.$$

1. Show that $f^{-1}\left(N\right)$ is a subgroup of $G.$
2. Show that $f^{-1}\left(N\right)\supseteq \ker f=f^{-1}\left(1\right).$
3. Show that if $N$ is a normal subgroup of $H$ then $f^{-1}\left(N\right)$ is a normal subgroup of $G.$

$$\begin{array}{rcl}G& \stackrel{f}{\to }& H\\ f^{-1}\left(N\right)& \mapsto & N\\ \cup |& & \cup |\\ \ker f=f^{-1}\left(1\right)& \mapsto & \left(1\right).\end{array}$$

c)

1. Let $M$ be a subgroup of $G$ and show that $M\subseteq f^{-1}\left(f\left(M\right)\right).$
2. Give an example of a homomorphism $f:G\to H$ and a subgroup $M$ of $G$ such that $M\ne f^{-1}\left(f\left(M\right)\right).$
3. Show that if $M\subseteq G$ is a subgroup of $G$ that contains $\ker f$ then $M=f^{-1}\left(f\left(M\right)\right).$

d)

1. Let $N$ be a subgroup of $H$ and show that $N\supseteq f\left(f^{-1}\left(N\right)\right).$
2. Give an example of a homomorphism $f:G\to H$ and a subgroup $N$ of $H$ such that $N\ne f\left(f^{-1}\left(N\right)\right).$
3. Show that if $N\subseteq H$ is a subgroup of $H$ and $N\subseteq \mathrm{im}f$ then $N=f^{-1}\left(f\left(N\right)\right).$

e)

1. Conclude from c) and d) that there is a one-to-one correspondence between subgroups of $G$ that contain $\ker f$ and subgroups of $H$ that are contained in $\mathrm{im}f.$
2. Give an example to show that this correspondence does not necessarily take normal subgroups to mormal subgroups.
3. Show that if $f$ is surjective then this correspondence takes normal subgroups of $G$ to normal subgroups of $H.$

6.

1. Let $H$ be a subgroup of a group $G.$ The canonical injection is the map $i:H\to G$ given by $$\begin{array}{rcll}i:& G& \to & H\\ & h& \mapsto & h.\end{array}$$
Show that $i:H\to G$ is a well defined injective homomorphism.
2. Let $N$ be a normal subgroup of a group $G.$ The canonical surjection or canonical projection is the map given by $$\begin{array}{rcll}\pi :& G& \to & G/N\\ & g& \mapsto & gN.\end{array}$$
Show that $\pi :G\to G/N$ is a well defined surjective homomorphism and that $\mathrm{im}\pi =G/N$ and $\ker \pi =N.$
3. Let $M$ be a subgroup of $G.$ Show, using Ex 1.1.5 that
1. $M/N=\left\{mN|m\in M\right\}$ is a subgroup of $G/N.$
2. $M/N$ is a normal subgroup of $G/N$ if $M$ is a normal subgroup of $G.$
3. $M/N=\pi \left(M\right)$ and if $M$ contains $N$ then ${\pi }^{-1}\left(\pi \left(M\right)\right)=M.$
4. Conclude that there is a one-to-one correspondence between subgroups of $G$ containing $N$ and subgroups of $G/N.$
5. Show that this correspondence takes normal subgroups to normal subgroups.

7. An exact sequence $$\dots \to G_{i-1}\stackrel{f_{i-1}}{\to }{G}_i\stackrel{f_i}{\to }{G}_{i+1}\to \dots$$ is a sequence of groups and group homomorphisms $f_i:G_i\to G_{i+1}$ such that $$\ker f_i=\mathrm{im}{f}_{i-1}$$ for all $i.$

A short exact sequence is an exact sequence of the form $$\left(1\right)\to K\stackrel{g}{\to }G\stackrel{f}{\to }H\to \left(1\right).$$

1. Show that in the above hort exact seqience $g$is always injective and $fis\; always\; surjective.$
2. Let $f:G\to H$ be a group homomorphism and show that the sequence $$\left(1\right)\to \ker f\to G\to \mathrm{im}f\to \left(1\right)$$ is always exact.
3. Let $K$ be a normal subgroup of a group $G.$ Let $i:K\to G$ be the canonical injection and let $\pi :G\to G/K$ be the canonical surjection. Show that $$\left(1\right)\to K\stackrel{i}{\to }G\stackrel{\pi }{\to }G/K\to \left(1\right)$$is a short exact sequence.

8. Let $N$ be a normal subgroup of a group $G.$ Let $K$ be a normal subgroup of $G$ containing $N.$ Then, by Ex 1.1.6 c)ii), $K/N=\left\{kN|k\in K\right\}$ is a normal subgroup of $G/N.$ Let $\frac{G/N}{K/N}$ be the quotient group and let $${\pi }_2:G/N\to \frac{G/N}{K/N}$$ be the canonical projection.

Let ${\pi }_1:G\to G/N$ be the canonical projection so that $$\left({\pi }_1\circ {\pi }_2\right):G\stackrel{{\pi }_1}{\to }G/N\stackrel{{\pi }_2}{\to }\frac{G/N}{K/N}.$$

1. Show that $\mathrm{im}\left({\pi }_1\circ {\pi }_2\right)=\frac{G/N}{K/N}.$
2. Show that $\ker \left({\pi }_1\circ {\pi }_2\right)=K.$
3. Using Theorem 2.4 b) conclude that $G/K\simeq \frac{G/N}{K/N}$ as groups.

9.

1. Prove that if $H$ and $K$ are subgroups of a group $G$ then $$HK=\left\{hk|h\in H,k\in K\right\}$$ is a subgroup of $G$ if al least one of $H$ and $K$ is normal in $G.$
2. Prove that if $H$ and $K$ are subgroups of a group $G$ and $K$ is normal in $G$ then the subgroup $$⟨H,K⟩=HK.$$ Warning! Don't even think that $\left|H\right|\left|K\right|=\left|HK\right|.$
3. Give an example of subgroups $H$ and $K$ of a group $G$ such that $K$ is normal and $\left|H\right|\left|K\right|\ne \left|HK\right|.$

10. Let $K$ be a normal subgroup of a group $G$ and let $H$ be any subgroup of $G.$ Let $$\begin{array}{rcll}\pi :& H& \to & G/K\\ & h& \mapsto & hK\end{array}$$ be the restriction of the canonical surjection $\pi :G\to G/K$ to $H.$

1. Show that $\ker \pi =H\cap K.$
2. Show that $\mathrm{im}\pi =\frac{HK}{K}=\left\{hK|h\in H\right\}.$
3. Using Theorem 2.4 b), conclude that $\frac{H}{H\cap K}\simeq \frac{HK}{K}.$

11. Let $H_1$ and $H_2$ be subgroups of groups $G_1$ and $G_2$ respectively.

1. Show that $H_1\times H_2$ is a subgroup of $G_1\times G_2.$
2. Let ${\pi }_1:G_1\to G_1/H_1$ and ${\pi }_2:G_2\to G_2/H_2$ be the canonical projections. Define a map $$\begin{array}{rcll}\left({\pi }_1\times {\pi }_2\right):& G_1\times G_2& \to & G_1/H_1\times G_2/H_2\\ & \left(g_1,g_2\right)& \mapsto & \left(g_1{H}_1,g_2{H}_2\right).\end{array}$$ Show that ${\pi }_1\times {\pi }_2$ is a well defined surjective group homomorphism.
3. Show that $\ker \left({\pi }_1\times {\pi }_2\right)=H_1\times H_2.$
4. Using Theorem 2.4 b), conclude that $$\frac{G_1\times G_2}{H_1\times H_2}\simeq \frac{G_1}{H_1}\times \frac{G_2}{H_2}.$$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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