Improve my browser-detecting JavaScript

Any suggestions how to make this code better? I'm novice in coding...)) In this code I'm want to show some hint depending of browser type on semi-transparent overlay. I'm detecting browser type (I'm need - Firefox, Chrome, IE 9 and higher), create all nessesary elements, append this elements to a document.body, and attach events. To detect "document.ready" I'm use domready.js

// Perform browsers detection;
    var browser = {};
    (function() {
        browser.firefox = false;
        browser.chrome = false;
        browser.msie = false;

        var nAgt = navigator.userAgent;
        var verOffset;

        if ( (verOffset=nAgt.indexOf("MSIE"))!=-1 ) {
            // For IE we also need to check browser version, because hint will be shown only if version is 9 or higher;
            browser.version = parseInt( nAgt.substring(verOffset+5), 10 );
            if (browser.version >= 9 ) {
                browser.msie = true;
                browser.name = "msie";
                browser.hint = {
                    position: "fixed",
                    display: "none",
                    zIndex: "101",
                    bottom: "50px",
                    left: "35%"
                };
            }
        }
        else if ( (verOffset=nAgt.indexOf("Chrome"))!=-1 ) {
            browser.chrome = true;
            browser.name = "chrome";
            browser.hint = {
                position: "fixed",
                display: "none",
                zIndex: "101",
                bottom: "0",
                left: "0"
            };
        }
        else if ( (verOffset=nAgt.indexOf("Firefox"))!=-1 ) {
            browser.firefox = true;
            browser.name = "firefox";
            browser.hint = {
                position: "fixed",
                display: "none",
                zIndex: "101",
                top: "0",
                right: "10px"
            };
        }
    })();

    domready(function() {
        if (browser.firefox || browser.chrome || browser.msie) {
            // Create "overlay" element
            var overlay = document.createElement("div");
            overlay.setAttribute("id", "hintOverlay");
            var overlayStyles = {
              width: "100%",
              height: "100%",
              background: "gray",
              opacity: "0.75",
              zIndex: "100",
              position: "fixed",
              top: "0",
              left: "0",
              display: "none"
            };
            for(var s in overlayStyles){
                if (overlayStyles.hasOwnProperty(s)) {
                    overlay.style[s] = overlayStyles[s];
                }
            }
            overlay.addEventListener("click", function() {
                this.style.display = "none";
                document.getElementById("hintElem").style.display = "none";
            });
            document.body.appendChild(overlay);

            // Create "hint" element
            var hintElem = document.createElement("img");
            hintElem.setAttribute("src", "images/" + browser.name + ".png");
            hintElem.setAttribute("id", "hintElem");
            for(var s in browser.hint) {
                if (browser.hint.hasOwnProperty(s)) {
                    hintElem.style[s] = browser.hint[s];
                }
            }
            document.body.appendChild(hintElem);

            // Attach click event to all links that have "download" in name
            var links = document.getElementsByTagName("a");
            for(var i = 0; i < links.length; i++) {
                if(links[i].getAttribute("id").indexOf('download') == 0) {
                    links[i].addEventListener("click", function() {
                        document.getElementById("hintOverlay").style.display = "block";
                        document.getElementById("hintElem").style.display = "block";
                    });
                }
            }
        }
    });