Adrian Parvin D. Ouano adrianparvino

A homomorphism whose kernel is trivial is injective. Proof: Given \(f(x) = f(y)\), then \(f(xy^-1) = f(x)f(y^-1) = e’\) Therefore \(xy^-1 = e\) and \(x = y\)

Notes for page 26

A kernel of f is the set \[\{ x ∈ G | f(x) = e’ \}\] An injective homomorphism \(f:G → G’\) is called an embedding.

A homomorphism whose kernel is trivial is injective.

Proof:

Let G be a group and S a subset of G. We shall say that S generates G, or that S is a set of generators for G, if every element of G can be expressed as a product of elements of S or inverses of elements of S, i.e. as a product x_1 \cdots x_n, where each x_i or x_i^-1 is in S. It is clear that the set of all such products is a subgroup of G, and is the smallest subgroup of G containing S. Thus S generates G if and only if the smallest subgroup of G containing S is G itself. If G is generated by S, then we write G = \langle S \rangle

	#define safe_init() safeenv *res; _safe_init(senv)
	#define safe_add(x, y, z) senv = _safe_add(senv, x, y, z); if (*senv == ERANGE); goto _safe_end;
	#define safe_end() _safe_end: free(senv)

	int foo ()
	{
	int a, b, c, d;

	safe_init();
	safe_add(a, b, &d);

	int foo ()
	{
	int a, b, c, d;

	T *res;
	res = safe_add(NULL, a, b, &d);
	res = safe_add(res , d, c, &d);
	}

	;; This buffer is for text that is not saved, and for Lisp evaluation.
	;; To create a file, visit it with <open> and enter text in its buffer.

	xserver = {
	enable = true;
	windowManager.i3.enable = true;
	desktopManager.xfce.enable = true;
	displayManager.lightdm.enable = true;
	};

	# Edit this configuration file to define what should be installed on
	# your system. Help is available in the configuration.nix(5) man page
	# and in the NixOS manual (accessible by running ‘nixos-help’).

	{ config, pkgs, lib, ... }:

	{
	imports =
	[ # Include the results of the hardware scan.
	./hardware-configuration.nix

	\documentclass{article}
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	\begin{tabular}{@{\hspace{0pt}}m{0.5\textwidth}@{\hspace{0pt}}>{\raggedleft}m{0.5\textwidth}@{\hspace{0pt}}}
	The quick & {The \\
	quick \\
	brown \\}
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